Introduction In this article, we will explore the concept of interval of convergence for a given power series. The power series in question is β n = 1 β ( β 1 ) n x 2 n n ! \sum_{n=1}^{\infty} \frac{(-1)^n x^{2n}}{n!} β n = 1 β β n ! ( β 1 ) n x 2 n β
The Ratio Test The ratio test is a method used to determine the convergence of a series. It is defined as follows:
Let β n = 1 β a n \sum_{n=1}^{\infty} a_n β n = 1 β β a n β β£ a n + 1 a n β£ \left|\frac{a_{n+1}}{a_n}\right| β a n β a n + 1 β β β n n n
Applying the Ratio Test to the Series We will now apply the ratio test to the series β n = 1 β ( β 1 ) n x 2 n n ! \sum_{n=1}^{\infty} \frac{(-1)^n x^{2n}}{n!} β n = 1 β β n ! ( β 1 ) n x 2 n β
Let a n = ( β 1 ) n x 2 n n ! a_n = \frac{(-1)^n x^{2n}}{n!} a n β = n ! ( β 1 ) n x 2 n β
β£ a n + 1 a n β£ = β£ ( β 1 ) n + 1 x 2 ( n + 1 ) ( n + 1 ) ! ( β 1 ) n x 2 n n ! β£ \left|\frac{a_{n+1}}{a_n}\right| = \left|\frac{\frac{(-1)^{n+1} x^{2(n+1)}}{(n+1)!}}{\frac{(-1)^n x^{2n}}{n!}}\right|
β a n β a n + 1 β β β = β n ! ( β 1 ) n x 2 n β ( n + 1 )! ( β 1 ) n + 1 x 2 ( n + 1 ) β β β
Simplifying the expression, we get:
β£ a n + 1 a n β£ = β£ ( β 1 ) n + 1 x 2 ( n + 1 ) ( n + 1 ) ! β
n ! ( β 1 ) n x 2 n β£ \left|\frac{a_{n+1}}{a_n}\right| = \left|\frac{(-1)^{n+1} x^{2(n+1)}}{(n+1)!} \cdot \frac{n!}{(-1)^n x^{2n}}\right|
β a n β a n + 1 β β β = β ( n + 1 )! ( β 1 ) n + 1 x 2 ( n + 1 ) β β
( β 1 ) n x 2 n n ! β β
β£ a n + 1 a n β£ = β£ ( β 1 ) n + 1 x 2 ( n + 1 ) ( n + 1 ) ! β
n ! ( β 1 ) n x 2 n β£ \left|\frac{a_{n+1}}{a_n}\right| = \left|\frac{(-1)^{n+1} x^{2(n+1)}}{(n+1)!} \cdot \frac{n!}{(-1)^n x^{2n}}\right|
β a n β a n + 1 β β β = β ( n + 1 )! ( β 1 ) n + 1 x 2 ( n + 1 ) β β
( β 1 ) n x 2 n n ! β β
β£ a n + 1 a n β£ = β£ ( β 1 ) x 2 n + 2 ( n + 1 ) n ! β
n ! ( β 1 ) n x 2 n β£ \left|\frac{a_{n+1}}{a_n}\right| = \left|\frac{(-1) x^{2n+2}}{(n+1)n!} \cdot \frac{n!}{(-1)^n x^{2n}}\right|
β a n β a n + 1 β β β = β ( n + 1 ) n ! ( β 1 ) x 2 n + 2 β β
( β 1 ) n x 2 n n ! β β
β£ a n + 1 a n β£ = β£ ( β 1 ) x 2 n + 2 ( n + 1 ) n ! β
n ! ( β 1 ) n x 2 n β£ \left|\frac{a_{n+1}}{a_n}\right| = \left|\frac{(-1) x^{2n+2}}{(n+1)n!} \cdot \frac{n!}{(-1)^n x^{2n}}\right|
β a n β a n + 1 β β β = β ( n + 1 ) n ! ( β 1 ) x 2 n + 2 β β
( β 1 ) n x 2 n n ! β β
β£ a n + 1 a n β£ = β£ ( β 1 ) x 2 n + 2 ( n + 1 ) n ! β
n ! ( β 1 ) n x 2 n β£ \left|\frac{a_{n+1}}{a_n}\right| = \left|\frac{(-1) x^{2n+2}}{(n+1)n!} \cdot \frac{n!}{(-1)^n x^{2n}}\right|
β a n β a n + 1 β β β = β ( n + 1 ) n ! ( β 1 ) x 2 n + 2 β β
( β 1 ) n x 2 n n ! β β
β£ a n + 1 a n β£ = β£ ( β 1 ) x 2 n + 2 ( n + 1 ) n ! β
n ! ( β 1 ) n x 2 n β£ \left|\frac{a_{n+1}}{a_n}\right| = \left|\frac{(-1) x^{2n+2}}{(n+1)n!} \cdot \frac{n!}{(-1)^n x^{2n}}\right|
β a n β a n + 1 β β β = β ( n + 1 ) n ! ( β 1 ) x 2 n + 2 β β
( β 1 ) n x 2 n n ! β β
β£ a n + 1 a n β£ = β£ ( β 1 ) x 2 n + 2 ( n + 1 ) n ! β
n ! ( β 1 ) n x 2 n β£ \left|\frac{a_{n+1}}{a_n}\right| = \left|\frac{(-1) x^{2n+2}}{(n+1)n!} \cdot \frac{n!}{(-1)^n x^{2n}}\right|
β a n β a n + 1 β β β = β ( n + 1 ) n ! ( β 1 ) x 2 n + 2 β β
( β 1 ) n x 2 n n ! β β
β£ a n + 1 a n β£ = β£ ( β 1 ) x 2 n + 2 ( n + 1 ) n ! β
n ! ( β 1 ) n x 2 n β£ \left|\frac{a_{n+1}}{a_n}\right| = \left|\frac{(-1) x^{2n+2}}{(n+1)n!} \cdot \frac{n!}{(-1)^n x^{2n}}\right|
β a n β a n + 1 β β β = β ( n + 1 ) n ! ( β 1 ) x 2 n + 2 β β
( β 1 ) n x 2 n n ! β β
β£ a n + 1 a n β£ = β£ ( β 1 ) x 2 n + 2 ( n + 1 ) n ! β
n ! ( β 1 ) n x 2 n β£ \left|\frac{a_{n+1}}{a_n}\right| = \left|\frac{(-1) x^{2n+2}}{(n+1)n!} \cdot \frac{n!}{(-1)^n x^{2n}}\right|
β a n β a n + 1 β β β = β ( n + 1 ) n ! ( β 1 ) x 2 n + 2 β β
( β 1 ) n x 2 n n ! β β
β£ a n + 1 a n β£ = β£ ( β 1 ) x 2 n + 2 ( n + 1 ) n ! β
n ! ( β 1 ) n x 2 n β£ \left|\frac{a_{n+1}}{a_n}\right| = \left|\frac{(-1) x^{2n+2}}{(n+1)n!} \cdot \frac{n!}{(-1)^n x^{2n}}\right|
β a n β a n + 1 β β β = β ( n + 1 ) n ! ( β 1 ) x 2 n + 2 β β
( β 1 ) n x 2 n n ! β β
β£ a n + 1 a n β£ = β£ ( β 1 ) x 2 n + 2 ( n + 1 ) n ! β
n ! ( β 1 ) n x 2 n β£ \left|\frac{a_{n+1}}{a_n}\right| = \left|\frac{(-1) x^{2n+2}}{(n+1)n!} \cdot \frac{n!}{(-1)^n x^{2n}}\right|
β a n β a n + 1 β β β = β ( n + 1 ) n ! ( β 1 ) x 2 n + 2 β β
( β 1 ) n x 2 n n ! β β
β£ a n + 1 a n β£ = β£ ( β 1 ) x 2 n + 2 ( n + 1 ) n ! β
n ! ( β 1 ) n x 2 n β£ \left|\frac{a_{n+1}}{a_n}\right| = \left|\frac{(-1) x^{2n+2}}{(n+1)n!} \cdot \frac{n!}{(-1)^n x^{2n}}\right|
β a n β a n + 1 β β β = β ( n + 1 ) n ! ( β 1 ) x 2 n + 2 β β
( β 1 ) n x 2 n n ! β β
β£ a n + 1 a n β£ = β£ ( β 1 ) x 2 n + 2 ( n + 1 ) n ! β
n ! ( β 1 ) n x 2 n β£ \left|\frac{a_{n+1}}{a_n}\right| = \left|\frac{(-1) x^{2n+2}}{(n+1)n!} \cdot \frac{n!}{(-1)^n x^{2n}}\right|
β a n β a n + 1 β β β = β ( n + 1 ) n ! ( β 1 ) x 2 n + 2 β β
( β 1 ) n x 2 n n ! β β
β£ a n + 1 a n β£ = β£ ( β 1 ) x 2 n + 2 ( n + 1 ) n ! β
n ! ( β 1 ) n x 2 n β£ \left|\frac{a_{n+1}}{a_n}\right| = \left|\frac{(-1) x^{2n+2}}{(n+1)n!} \cdot \frac{n!}{(-1)^n x^{2n}}\right|
β a n β a n + 1 β β β = β ( n + 1 ) n ! ( β 1 ) x 2 n + 2 β β
( β 1 ) n x 2 n n ! β β
β£ a n + 1 a n β£ = β£ ( β 1 ) x 2 n + 2 ( n + 1 ) n ! β
n ! ( β 1 ) n x 2 n β£ \left|\frac{a_{n+1}}{a_n}\right| = \left|\frac{(-1) x^{2n+2}}{(n+1)n!} \cdot \frac{n!}{(-1)^n x^{2n}}\right|
β a n β a n + 1 β β β = β ( n + 1 ) n ! ( β 1 ) x 2 n + 2 β β
( β 1 ) n x 2 n n ! β β
β£ a n + 1 a n β£ = β£ ( β 1 ) x 2 n + 2 ( n + 1 ) n ! β
n ! ( β 1 ) n x 2 n β£ \left|\frac{a_{n+1}}{a_n}\right| = \left|\frac{(-1) x^{2n+2}}{(n+1)n!} \cdot \frac{n!}{(-1)^n x^{2n}}\right|
β a n β a n + 1 β β β = β ( n + 1 ) n ! ( β 1 ) x 2 n + 2 β β
( β 1 ) n x 2 n n ! β β
β£ a n + 1 a n β£ = β£ ( β 1 ) x 2 n + 2 ( n + 1 ) n ! β
n ! ( β 1 ) n x 2 n β£ \left|\frac{a_{n+1}}{a_n}\right| = \left|\frac{(-1) x^{2n+2}}{(n+1)n!} \cdot \frac{n!}{(-1)^n x^{2n}}\right|
β a n β a n + 1 β β β = β ( n + 1 ) n ! ( β 1 ) x 2 n + 2 β β
( β 1 ) n x 2 n n ! β β
β£ a n + 1 a n β£ = β£ ( β 1 ) x 2 n + 2 ( n + 1 ) n ! β
n ! ( β 1 ) n x 2 n β£ \left|\frac{a_{n+1}}{a_n}\right| = \left|\frac{(-1) x^{2n+2}}{(n+1)n!} \cdot \frac{n!}{(-1)^n x^{2n}}\right|
β a n β a n + 1 β β β = β ( n + 1 ) n ! ( β 1 ) x 2 n + 2 β β
( β 1 ) n x 2 n n ! β β
β£ a n + 1 a n β£ = β£ ( β 1 ) x 2 n + 2 ( n + 1 ) n ! β
n ! ( β 1 ) n x 2 n β£ \left|\frac{a_{n+1}}{a_n}\right| = \left|\frac{(-1) x^{2n+2}}{(n+1)n!} \cdot \frac{n!}{(-1)^n x^{2n}}\right|
β a n β a n + 1 β β β = β ( n + 1 ) n ! ( β 1 ) x 2 n + 2 β β
( β 1 ) n x 2 n n ! β β
β£ a n + 1 a n β£ = β£ ( β 1 ) x 2 n + 2 ( n + 1 ) n ! β
n ! ( β 1 ) n x 2 n β£ \left|\frac{a_{n+1}}{a_n}\right| = \left|\frac{(-1) x^{2n+2}}{(n+1)n!} \cdot \frac{n!}{(-1)^n x^{2n}}\right|
β a n β a n + 1 β β β = β ( n + 1 ) n ! ( β 1 ) x 2 n + 2 β β
( β 1 ) n x 2 n n ! β β
β£ a n + 1 a n β£ = β£ ( β 1 ) x 2 n + 2 ( n + 1 ) n ! β
n ! ( β 1 ) n x 2 n β£ \left|\frac{a_{n+1}}{a_n}\right| = \left|\frac{(-1) x^{2n+2}}{(n+1)n!} \cdot \frac{n!}{(-1)^n x^{2n}}\right|
β a n β a n + 1 β β β = β ( n + 1 ) n ! ( β 1 ) x 2 n + 2 β β
( β 1 ) n x 2 n n ! β β
β£ a n + 1 a n β£ = β£ ( β 1 ) x 2 n + 2 ( n + 1 ) n ! β
n ! ( β 1 ) n x 2 n β£ \left|\frac{a_{n+1}}{a_n}\right| = \left|\frac{(-1) x^{2n+2}}{(n+1)n!} \cdot \frac{n!}{(-1)^n x^{2n}}\right|
β a n β a n + 1 β β β = β ( n + 1 ) n ! ( β 1 ) x 2 n + 2 β β
( β 1 ) n x 2 n n ! β β
β£ a n + 1 a n β£ = β£ ( β 1 ) x 2 n + 2 ( n + 1 ) n ! β
n ! ( β 1 ) n x 2 n β£ \left|\frac{a_{n+1}}{a_n}\right| = \left|\frac{(-1) x^{2n+2}}{(n+1)n!} \cdot \frac{n!}{(-1)^n x^{2n}}\right|
β a n β a n + 1 β β β = β ( n + 1 ) n ! ( β 1 ) x 2 n + 2 β β
( β 1 ) n x 2 n n ! β β
β£ a n + 1 a n β£ = β£ ( β 1 ) x 2 n + 2 ( n + 1 ) n ! β
n ! ( β 1 ) n x 2 n β£ \left|\frac{a_{n+1}}{a_n}\right| = \left|\frac{(-1) x^{2n+2}}{(n+1)n!} \cdot \frac{n!}{(-1)^n x^{2n}}\right|
β a n β a n + 1 β β β = β ( n + 1 ) n ! ( β 1 ) x 2 n + 2 β β
( β 1 ) n x 2 n n ! β β
β£ a n + 1 a n β£ = β£ ( β 1 ) x 2 n + 2 ( n + 1 ) n ! β
n ! ( β 1 ) n x 2 n β£ \left|\frac{a_{n+1}}{a_n}\right| = \left|\frac{(-1) x^{2n+2}}{(n+1)n!} \cdot \frac{n!}{(-1)^n x^{2n}}\right|
β a n β a n + 1 β β β = β ( n + 1 ) n ! ( β 1 ) x 2 n + 2 β β
( β 1 ) n x 2 n n ! β β
β£ a n + 1 a n β£ = β£ ( β 1 ) x 2 n + 2 ( n + 1 ) n ! β
n ! ( β 1 ) n x 2 n β£ \left|\frac{a_{n+1}}{a_n}\right| = \left|\frac{(-1) x^{2n+2}}{(n+1)n!} \cdot \frac{n!}{(-1)^n x^{2n}}\right|
β a n β a n + 1 β β β = β ( n + 1 ) n ! ( β 1 ) x 2 n + 2 β β
( β 1 ) n x 2 n n ! β β
\left|\frac{a_{n+1}}{a_n}\right| = \left|\frac{(-1) x^{2n+2<br/>
**Determine the Interval of Convergence for the Series $\sum_{n=1}^{\infty} \frac{(-1)^n x^{2n}}{n!}$**
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**Q&A**
------
**Q: What is the interval of convergence for the series $\sum_{n=1}^{\infty} \frac{(-1)^n x^{2n}}{n!}$?**
A: To determine the interval of convergence, we need to find the values of $x$ for which the series converges.
**Q: How do we determine the interval of convergence?**
A: We can use the ratio test to determine the interval of convergence. The ratio test states that if the limit of $\left|\frac{a_{n+1}}{a_n}\right|$ as $n$ approaches infinity is less than 1, the series converges. If the limit is greater than 1, the series diverges. If the limit is equal to 1, the test is inconclusive.
**Q: What is the ratio test?**
A: The ratio test is a method used to determine the convergence of a series. It is defined as follows:
Let $\sum_{n=1}^{\infty} a_n$ be a series. Then, the ratio test states that if the limit of $\left|\frac{a_{n+1}}{a_n}\right|$ as $n$ approaches infinity is less than 1, the series converges. If the limit is greater than 1, the series diverges. If the limit is equal to 1, the test is inconclusive.
**Q: How do we apply the ratio test to the series $\sum_{n=1}^{\infty} \frac{(-1)^n x^{2n}}{n!}$?**
A: We need to find the limit of $\left|\frac{a_{n+1}}{a_n}\right|$ as $n$ approaches infinity.
Let $a_n = \frac{(-1)^n x^{2n}}{n!}$. Then, we have:
$\left|\frac{a_{n+1}}{a_n}\right| = \left|\frac{\frac{(-1)^{n+1} x^{2(n+1)}}{(n+1)!}}{\frac{(-1)^n x^{2n}}{n!}}\right|
Simplifying the expression, we get:
β£ a n + 1 a n β£ = β£ ( β 1 ) n + 1 x 2 ( n + 1 ) ( n + 1 ) ! β
n ! ( β 1 ) n x 2 n β£ \left|\frac{a_{n+1}}{a_n}\right| = \left|\frac{(-1)^{n+1} x^{2(n+1)}}{(n+1)!} \cdot \frac{n!}{(-1)^n x^{2n}}\right|
β a n β a n + 1 β β β = β ( n + 1 )! ( β 1 ) n + 1 x 2 ( n + 1 ) β β
( β 1 ) n x 2 n n ! β β
β£ a n + 1 a n β£ = β£ ( β 1 ) x 2 n + 2 ( n + 1 ) n ! β
n ! ( β 1 ) n x 2 n β£ \left|\frac{a_{n+1}}{a_n}\right| = \left|\frac{(-1) x^{2n+2}}{(n+1)n!} \cdot \frac{n!}{(-1)^n x^{2n}}\right|
β a n β a n + 1 β β β = β ( n + 1 ) n ! ( β 1 ) x 2 n + 2 β β
( β 1 ) n x 2 n n ! β β
β£ a n + 1 a n β£ = β£ ( β 1 ) x 2 n + 2 ( n + 1 ) n ! β
n ! ( β 1 ) n x 2 n β£ \left|\frac{a_{n+1}}{a_n}\right| = \left|\frac{(-1) x^{2n+2}}{(n+1)n!} \cdot \frac{n!}{(-1)^n x^{2n}}\right|
β a n β a n + 1 β β β = β ( n + 1 ) n ! ( β 1 ) x 2 n + 2 β β
( β 1 ) n x 2 n n ! β β
β£ a n + 1 a n β£ = β£ ( β 1 ) x 2 n + 2 ( n + 1 ) n ! β
n ! ( β 1 ) n x 2 n β£ \left|\frac{a_{n+1}}{a_n}\right| = \left|\frac{(-1) x^{2n+2}}{(n+1)n!} \cdot \frac{n!}{(-1)^n x^{2n}}\right|
β a n β a n + 1 β β β = β ( n + 1 ) n ! ( β 1 ) x 2 n + 2 β β
( β 1 ) n x 2 n n ! β β
β£ a n + 1 a n β£ = β£ ( β 1 ) x 2 n + 2 ( n + 1 ) n ! β
n ! ( β 1 ) n x 2 n β£ \left|\frac{a_{n+1}}{a_n}\right| = \left|\frac{(-1) x^{2n+2}}{(n+1)n!} \cdot \frac{n!}{(-1)^n x^{2n}}\right|
β a n β a n + 1 β β β = β ( n + 1 ) n ! ( β 1 ) x 2 n + 2 β β
( β 1 ) n x 2 n n ! β β
β£ a n + 1 a n β£ = β£ ( β 1 ) x 2 n + 2 ( n + 1 ) n ! β
n ! ( β 1 ) n x 2 n β£ \left|\frac{a_{n+1}}{a_n}\right| = \left|\frac{(-1) x^{2n+2}}{(n+1)n!} \cdot \frac{n!}{(-1)^n x^{2n}}\right|
β a n β a n + 1 β β β = β ( n + 1 ) n ! ( β 1 ) x 2 n + 2 β β
( β 1 ) n x 2 n n ! β β
β£ a n + 1 a n β£ = β£ ( β 1 ) x 2 n + 2 ( n + 1 ) n ! β
n ! ( β 1 ) n x 2 n β£ \left|\frac{a_{n+1}}{a_n}\right| = \left|\frac{(-1) x^{2n+2}}{(n+1)n!} \cdot \frac{n!}{(-1)^n x^{2n}}\right|
β a n β a n + 1 β β β = β ( n + 1 ) n ! ( β 1 ) x 2 n + 2 β β
( β 1 ) n x 2 n n ! β β
β£ a n + 1 a n β£ = β£ ( β 1 ) x 2 n + 2 ( n + 1 ) n ! β
n ! ( β 1 ) n x 2 n β£ \left|\frac{a_{n+1}}{a_n}\right| = \left|\frac{(-1) x^{2n+2}}{(n+1)n!} \cdot \frac{n!}{(-1)^n x^{2n}}\right|
β a n β a n + 1 β β β = β ( n + 1 ) n ! ( β 1 ) x 2 n + 2 β β
( β 1 ) n x 2 n n ! β β
β£ a n + 1 a n β£ = β£ ( β 1 ) x 2 n + 2 ( n + 1 ) n ! β
n ! ( β 1 ) n x 2 n β£ \left|\frac{a_{n+1}}{a_n}\right| = \left|\frac{(-1) x^{2n+2}}{(n+1)n!} \cdot \frac{n!}{(-1)^n x^{2n}}\right|
β a n β a n + 1 β β β = β ( n + 1 ) n ! ( β 1 ) x 2 n + 2 β β
( β 1 ) n x 2 n n ! β β
β£ a n + 1 a n β£ = β£ ( β 1 ) x 2 n + 2 ( n + 1 ) n ! β
n ! ( β 1 ) n x 2 n β£ \left|\frac{a_{n+1}}{a_n}\right| = \left|\frac{(-1) x^{2n+2}}{(n+1)n!} \cdot \frac{n!}{(-1)^n x^{2n}}\right|
β a n β a n + 1 β β β = β ( n + 1 ) n ! ( β 1 ) x 2 n + 2 β β
( β 1 ) n x 2 n n ! β β
β£ a n + 1 a n β£ = β£ ( β 1 ) x 2 n + 2 ( n + 1 ) n ! β
n ! ( β 1 ) n x 2 n β£ \left|\frac{a_{n+1}}{a_n}\right| = \left|\frac{(-1) x^{2n+2}}{(n+1)n!} \cdot \frac{n!}{(-1)^n x^{2n}}\right|
β a n β a n + 1 β β β = β ( n + 1 ) n ! ( β 1 ) x 2 n + 2 β β
( β 1 ) n x 2 n n ! β β
β£ a n + 1 a n β£ = β£ ( β 1 ) x 2 n + 2 ( n + 1 ) n ! β
n ! ( β 1 ) n x 2 n β£ \left|\frac{a_{n+1}}{a_n}\right| = \left|\frac{(-1) x^{2n+2}}{(n+1)n!} \cdot \frac{n!}{(-1)^n x^{2n}}\right|
β a n β a n + 1 β β β = β ( n + 1 ) n ! ( β 1 ) x 2 n + 2 β β
( β 1 ) n x 2 n n ! β β
β£ a n + 1 a n β£ = β£ ( β 1 ) x 2 n + 2 ( n + 1 ) n ! β
n ! ( β 1 ) n x 2 n β£ \left|\frac{a_{n+1}}{a_n}\right| = \left|\frac{(-1) x^{2n+2}}{(n+1)n!} \cdot \frac{n!}{(-1)^n x^{2n}}\right|
β a n β a n + 1 β β β = β ( n + 1 ) n ! ( β 1 ) x 2 n + 2 β β
( β 1 ) n x 2 n n ! β β
β£ a n + 1 a n β£ = β£ ( β 1 ) x 2 n + 2 ( n + 1 ) n ! β
n ! ( β 1 ) n x 2 n β£ \left|\frac{a_{n+1}}{a_n}\right| = \left|\frac{(-1) x^{2n+2}}{(n+1)n!} \cdot \frac{n!}{(-1)^n x^{2n}}\right|
β a n β a n + 1 β β β = β ( n + 1 ) n ! ( β 1 ) x 2 n + 2 β β
( β 1 ) n x 2 n n ! β β
β£ a n + 1 a n β£ = β£ ( β 1 ) x 2 n + 2 ( n + 1 ) n ! β
n ! ( β 1 ) n x 2 n β£ \left|\frac{a_{n+1}}{a_n}\right| = \left|\frac{(-1) x^{2n+2}}{(n+1)n!} \cdot \frac{n!}{(-1)^n x^{2n}}\right|
β a n β a n + 1 β β β = β ( n + 1 ) n ! ( β 1 ) x 2 n + 2 β β
( β 1 ) n x 2 n n ! β β
β£ a n + 1 a n β£ = β£ ( β 1 ) x 2 n + 2 ( n + 1 ) n ! β
n ! ( β 1 ) n x 2 n β£ \left|\frac{a_{n+1}}{a_n}\right| = \left|\frac{(-1) x^{2n+2}}{(n+1)n!} \cdot \frac{n!}{(-1)^n x^{2n}}\right|
β a n β a n + 1 β β β = β ( n + 1 ) n ! ( β 1 ) x 2 n + 2 β β
( β 1 ) n x 2 n n ! β β
β£ a n + 1 a n β£ = β£ ( β 1 ) x 2 n + 2 ( n + 1 ) n ! β
n ! ( β 1 ) n x 2 n β£ \left|\frac{a_{n+1}}{a_n}\right| = \left|\frac{(-1) x^{2n+2}}{(n+1)n!} \cdot \frac{n!}{(-1)^n x^{2n}}\right|
β a n β a n + 1 β β β = β ( n + 1 ) n ! ( β 1 ) x 2 n + 2 β β
( β 1 ) n x 2 n n ! β β
β£ a n + 1 a n β£ = β£ ( β 1 ) x 2 n + 2 ( n + 1 ) n ! β
n ! ( β 1 ) n x 2 n β£ \left|\frac{a_{n+1}}{a_n}\right| = \left|\frac{(-1) x^{2n+2}}{(n+1)n!} \cdot \frac{n!}{(-1)^n x^{2n}}\right|
β a n β a n + 1 β β β = β ( n + 1 ) n ! ( β 1 ) x 2 n + 2 β β
( β 1 ) n x 2 n n ! β β
β£ a n + 1 a n β£ = β£ ( β 1 ) x 2 n + 2 ( n + 1 ) n ! β
n ! ( β 1 ) n x 2 n β£ \left|\frac{a_{n+1}}{a_n}\right| = \left|\frac{(-1) x^{2n+2}}{(n+1)n!} \cdot \frac{n!}{(-1)^n x^{2n}}\right|
β a n β a n + 1 β β β = β ( n + 1 ) n ! ( β 1 ) x 2 n + 2 β β
( β 1 ) n x 2 n n ! β β
β£ a n + 1 a n β£ = β£ ( β 1 ) x 2 n + 2 ( n + 1 ) n ! β
n ! ( β 1 ) n x 2 n β£ \left|\frac{a_{n+1}}{a_n}\right| = \left|\frac{(-1) x^{2n+2}}{(n+1)n!} \cdot \frac{n!}{(-1)^n x^{2n}}\right|
β a n β a n + 1 β β β = β ( n + 1 ) n ! ( β 1 ) x 2 n + 2 β β
( β 1 ) n x 2 n n ! β β
β£ a n + 1 a n β£ = β£ ( β 1 ) x 2 n + 2 ( n + 1 ) n ! β
n ! ( β 1 ) n x 2 n β£ \left|\frac{a_{n+1}}{a_n}\right| = \left|\frac{(-1) x^{2n+2}}{(n+1)n!} \cdot \frac{n!}{(-1)^n x^{2n}}\right|
β a n β a n + 1 β β β = β ( n + 1 ) n ! ( β 1 ) x 2 n + 2 β β
( β 1 ) n x 2 n n ! β β
β£ a n + 1 a n β£ = β£ ( β 1 ) x 2 n + 2 ( n + 1 ) n ! β
n ! ( β 1 ) n x 2 n β£ \left|\frac{a_{n+1}}{a_n}\right| = \left|\frac{(-1) x^{2n+2}}{(n+1)n!} \cdot \frac{n!}{(-1)^n x^{2n}}\right|
β a n β a n + 1 β β β = β ( n + 1 ) n ! ( β 1 ) x 2 n + 2 β β
( β 1 ) n x 2 n n ! β β
β£ a n + 1 a n β£ = β£ ( β 1 ) x 2 n + 2 ( n + 1 ) n ! β
n ! ( β 1 ) n x 2 n β£ \left|\frac{a_{n+1}}{a_n}\right| = \left|\frac{(-1) x^{2n+2}}{(n+1)n!} \cdot \frac{n!}{(-1)^n x^{2n}}\right|
β a n β a n + 1 β β β = β ( n + 1 ) n ! ( β 1 ) x 2 n + 2 β β
( β 1 ) n x 2 n n ! β β
β£ a n + 1 a n β£ = β£ ( β 1 ) x 2 n + 2 ( n + 1 ) n ! β
n ! ( β 1 ) n x 2 n β£ \left|\frac{a_{n+1}}{a_n}\right| = \left|\frac{(-1) x^{2n+2}}{(n+1)n!} \cdot \frac{n!}{(-1)^n x^{2n}}\right|
β a n β a n + 1 β β β = β ( n + 1 ) n ! ( β 1 ) x 2 n + 2 β β
( β 1 ) n x 2 n n ! β β
\left|\frac{a_{n+1}}{a_n}\right| = \left|\frac{(-1) x^{2n+2