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Telescoping series For the following
Chapter 11, Problem 51E(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.
\(\sum_{k=1}^{\infty} \ln \left(\frac{k+1}{k}\right)\)
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.
\(\sum_{k=1}^{\infty} \ln \left(\frac{k+1}{k}\right)\)
ANSWER:Problem 51ETelescoping 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.Solution:Step 1 ( Using )We observe that consecutive terms cancel each other.A formula for the nth term of the sequence of partial sums {Sn} is:.