Solution Found!
Telescoping series For the following
Chapter 11, Problem 54E(choose chapter or problem)
47-58. Telescoping series For the following telescoping series, find a formula for the nth term of the sequence of partial sums \(\left\{S_{n}\right\}\). Then evaluate \(\lim_{n\rightarrow\infty}\ S_n\), to obtain the value of the series or state that the series diverges.
\(\sum_{k=1}^{\infty} \frac{1}{(a k+1)(a k+a+1)}\), where a is a positive integer
Questions & Answers
QUESTION:
47-58. Telescoping series For the following telescoping series, find a formula for the nth term of the sequence of partial sums \(\left\{S_{n}\right\}\). Then evaluate \(\lim_{n\rightarrow\infty}\ S_n\), to obtain the value of the series or state that the series diverges.
\(\sum_{k=1}^{\infty} \frac{1}{(a k+1)(a k+a+1)}\), where a is a positive integer
ANSWER:Problem 54ETelescoping series For the following telescoping series, find a formula for the nth term of the sequence of partial sums {Sn}. Then evaluate to obtain the value of the series or stale that the series diverges. , where a is a positive integerAnswer ; Step-1 ; The given Telescoping series is , where a is a positive integer In this problem we have to find the formula for term in and then we have to evaluate or we have state that the series diverges.Consider = - ) , since - = = . = - ) ………..(1)Let us first find the term of the sequence of partial sums = - ) ……………(2)Substitute values for we get +[ - ]} Cancelling the like terms with opposite sign we get, = { } = - = = = Thus the term in the series is =