Chapter 8: Q 49. (page 670)

Find the radius of convergence for the given series: $\sum_{k = 0}^{\infty} \frac{k!}{(k + m)!} x^{k}$

Short Answer

Expert verified

The radius of convergence for the series is $1$ .

Step by step solution

Step 1. Given information.

The given power series is $\sum_{k = 0}^{\infty} \frac{k!}{(k + m)!} x$ .

Step 2. Find the radius of convergence.

Let us take $b_{k} = \frac{k!}{(k + m)!} x^{k}$ therefore $b_{k + 1} = \frac{(k + 1)!}{(k + 1 + m)!} x^{k + 1}$

Ratio for the absolute convergence is

$\lim_{k \to \infty} |\frac{b_{k + 1}}{b_{k}}| = \lim_{k \to \infty} |\frac{\frac{(k + 1)!}{(k + 1 + m)!} x^{k + 1}}{\frac{k!}{(k + m)!} x^{k}}| = \lim_{k \to \infty} |(\frac{k + 1}{k + 1 + m}) x|$

So, by the ratio test of absolute convergence, the series will converge when $|x| < 1$ .

This implies that $x \in (- 1, 1)$ .

Therefore, the radius of the convergence for the series is $1$ .

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Find the radius of convergence for the given series: $\sum_{k = 0}^{\infty} \frac{k!}{(k + m)!} x^{k}$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Find the radius of convergence.

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Find the radius of convergence for the given series:∑k=0∞k!k+m!xk

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Find the radius of convergence.

One App. One Place for Learning.

Most popular questions from this chapter

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Find the radius of convergence for the given series: $\sum_{k = 0}^{\infty} \frac{k!}{(k + m)!} x^{k}$