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Solved: Convergence parameter Find the values of the
Chapter 12, Problem 63E(choose chapter or problem)
58-65. Convergence parameter Find the values of the parameter p for which the following series converge.
\(\sum_{k=1}^{\infty} \frac{1 \cdot 3 \cdot 5 \cdots(2 k-1)}{k p^{k+1} k !}\)
Questions & Answers
QUESTION:
58-65. Convergence parameter Find the values of the parameter p for which the following series converge.
\(\sum_{k=1}^{\infty} \frac{1 \cdot 3 \cdot 5 \cdots(2 k-1)}{k p^{k+1} k !}\)
ANSWER:Problem 63E
Convergence parameter Find the values of the parameter p for which the following series converge.
Solution
Step 1
In this problem we have to find the value of parameter p for which the series converge.
Since the series has a factorial term, let us use ratio test.
Ratio test:
Let be a sequence of nonzero terms and let . Then
- If L < 1 then is convergent.
- If L > 1 then is divergent.
- If L = 1 then nothing can be said about the series. In other words, we say that the ratio is inconclusive.