Explore BrainMass

# radius of the power series

Not what you're looking for? Search our solutions OR ask your own Custom question.

This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

Find the radius of convergence of ∑▒〖(a_n x^n)〗, given a_n=1when n is the square of a natural number and a_n otherwise. If a_n=1 when n=m! for n∈N and a_n=0 otherwise, find the radius of convergence of ∑▒〖(a_n x^n)〗.

Determine the radius of convergence of ∑▒〖(a_n x^n)〗, if 0<ρ≤|a_n |≤q for all n∈N.

## SOLUTION This solution is FREE courtesy of BrainMass!

To find the radius of the power series , we use the following theorem.
Theorem: Let , then the radius of convergence is .

Problem #1
(a) Find the radius of convergence of , given if is the square of a natural number and otherwise. I think you missed in the posting.
From the condition, we know that , then we have

So the radius of convergence is .
(b) Find the radius of convergence of , given when and otherwise.
From the condition, we know that , then we have

So the radius of convergence is .

Problem #2
Find the radius of convergence of , if for all .
From the condition, we know that , then we have

Here we use the fact that for any positive number , we have . This is because as .
Thus we must have

So the radius of convergence is .

This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!