Chapter 8: Q 51. (page 670)

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

Short Answer

Expert verified

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

Step by step solution

Step 1. Given information.

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

Step 2. Find the radius of convergence.

Let us take $b_{k} = \frac{k!}{(k + m)! (k + m)!} x^{k}$ therefore localid="1649443626355" $b_{k + 1} = \frac{(k + 1)!}{(k + 1 + m)! (k + 1 + m)!} x^{k + 1}$

Ratio for the absolute convergence is

localid="1649443468258" $\lim_{k \to \infty} |\frac{b_{k + 1}}{b_{k}}| = \lim_{k \to \infty} |\frac{\frac{(k + 1)!}{(k + 1 + m)! (k + 1 + m)!} x^{k + 1}}{\frac{k!}{(k + m)! (k + m)!} x^{k}}| = \lim_{k \to \infty} |(\frac{k + 1}{(k + m) (k + m)}) x|$

Since, for $k \to \infty$ The limit is zero irrespective of the value of variable.

$\lim_{k \to \infty} (\frac{k + 1}{(k + m) (k + m)}) |x| = 0$

So, by the ratio test the series converges absolutely for every value of $x$ .

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

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

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

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

Recommended explanations on Math Textbooks

Mechanics Maths

Probability and Statistics

Applied Mathematics

Logic and Functions

Pure Maths

Calculus

Study anywhere. Anytime. Across all devices.

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

Recommended explanations on Math Textbooks

Mechanics Maths

Probability and Statistics

Applied Mathematics

Logic and Functions

Pure Maths

Calculus

Study anywhere. Anytime. Across all devices.

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