Test the following series for convergence,

(i) 

(ii) 

Part (i)
Consider the series
\[
\sum_{n=1}^{\infty} n \, x^{n-1}.
\]

We can recognize this as a geometric series, with each term being \(n \, x^{n-1}\). By manipulating the original series, we can derive a formula to test its convergence.

Given a geometric series of the form \(\sum_{n=0}^{\infty} a r^n\), it converges if \(|r| < 1\). Similarly, the series provided can be rewritten in a simpler form by noting that it is an infinite series starting at \(n ______ ___ ___ _____ ______.
_____ __________ _____ ______ ____ __________ ______.
____ __________ ________ _____ _____ _________ _______ _________ _______ _________ _______ ________.
____ ________ _____ ________ ______ ____.
________ _________ ______ _____ _______ ____ _________ _________ ____ _______ ____.
___ ________ ____ ________ ______ __________ ________.
___ _____ ___ _________ ______ ___ ____ ________ _________ __________ ___.
_____ ____ _____ __________ _________ _____ ______ __________ ___ _______ _______.
__________ ________ __________ ____ _____ _____ ________ __________ _____ ______ ___.
_________ _______ ________ ____ ______ ____.
________ ______ _______ ________ ____.
__________ ___ _______ ___ ___ ___ ______.
__________ ______ ________ _________ ___ _______.
__________ ________ ___ _______ _______ ___.
______ ______ __________ ______ _________.
____ __________ ________ _______ ____ _______ _________ ______ _____ ____.
__________ _______ ________ ________ _________ _______ __________ __________.
________ _____ ________ __________ ______ _________ ________ _________.
_____ __________ __________ __________ _______ __________ _______ _______.
________ _________ _________ ______ ___ _____.
_______ ___ _______ _____ _____ _________ __________ ______.
________ _______ _____ _________ __________ ________ ________ _____.
________ ______ ____ ____ ________ ____ _________ _______ ___.
__________ ________ ___ ___ ________.
______ _________ ___ ______ ___ _________ _______ ____.
_______ ______ ___ ______ ____ ________ _____ _____ ________ _____.
___ ____ ____ ____ ______ ____ ______.
____ __________ ________ _____ ____ ________ _______ ______ ________ _________ ______.
___ ____ ___ _________ ______ _______ __________ _______ ______ ________ ________.
_____ __________ ______ _________ _______.
________ _________ ______ _____ ______ __________ _______.
___ ______ ______ _________ ________ _______ ________ _____.
_______ __________ _______ ________ _____ ___ ______ _______ _________ __________.
______ _____ ________ ___ __________ _____ ___ ________ __________ ________.
________ __________ _______ _________ _________ __________ ________ __________ _______.
__________ __________ ___ _________ ___ _________ _________.
______ _____ ______ _________ _________ ______ __________.
__________ ____ _____ ________ _________ ________ _________.
__________ _____ ______ ___ ____ ____ __________ ____ _________ ________.
____ _________ __________ _____ _________ _________ ______.
_____ ________ ________ ______ ___ ___ __________ ___ _______ ________ _____ ___.
Get Full Answer on WhatsApp