Chapter 7: Q. 30 (page 657)

Check the convergence $\sum_{k = 0}^{\infty} \frac{5^{2 k + 1}}{(2 k + 1)!}$

Short Answer

Expert verified

The series converges

Step by step solution

Step 1. Given

The given series is $\sum_{k = 0}^{\infty} \frac{5^{2 k + 1}}{(2 k + 1)!}$ .

Step 2. Ratio test

$According to the RatioTest, {\sum a_{k}}_{k = 1}^{\infty} be the series with positive terms, if L = \lim_{k \to \infty} \frac{a_{k + 1}}{a_{k}} 1 . If L < 1 series converges . 2 . If L > 1 series diverges . 3 . If L = 1 the test is inconclusive .$

Step 3. Checking the convergence

$Now, calculate the value of \frac{a_{k + 1}}{a_{k}} Consider the general term a s = \frac{5^{2 k + 1}}{(2 k + 1)! .} . . . (1 T h e r e f o r e, b y s u b s t i t u t i n g k = k + 1 \ln (1) a_{k + 1} = \frac{5^{2 (k + 1) + 1}}{(2 (k + 1) + 1)!} = 5^{2 k + 3 (} 2 k + 3)! N o w, \frac{a_{k + 1}}{a_{k}} = \frac{\frac{5^{2 k + 3}}{(2 k + 3)}}{5^{2 k +!}} U \sin g f o r m u l a n! = n (n - 1)! \frac{a_{k + 1}}{a_{k}} = \frac{25}{(2 k + 3) (2 k + 2)} T a k i n g l i m i t s \lim_{k \to \infty} \frac{a_{k + 1}}{a_{k}} = \lim_{k \to \infty} (\frac{25}{k (2 + \frac{3}{k}) (2 + \frac{2}{k})}) L = 0$

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Check the convergence $\sum_{k = 0}^{\infty} \frac{5^{2 k + 1}}{(2 k + 1)!}$

Short Answer

Step by step solution

Step 1. Given

Step 2. Ratio test

Step 3. Checking the convergence

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Step by step solution

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Check the convergence $\sum_{k = 0}^{\infty} \frac{5^{2 k + 1}}{(2 k + 1)!}$