**Problem 6: Telescoping Series**Given the telescoping series:\[ \sum_{k=1}^{\infty} \frac{2}{(2k+3)(2k+5)} \]1. **Find a formula for the $n$th term of the sequence of partial sums $\{S_n\}$.**2. **Evaluate $ \lim_{n \to \infty} S_n $ to obtain the value of the series or state that the series diverges.****Explanation:**- A **telescoping series** is a series whose partial sums eventually only have a fixed number of terms after cancellation.- The given expression is in the form of a fraction, and we aim to simplify and break it into more manageable components for summation.- To achieve the solution, decompose the fraction and simplify the terms, observing the pattern in partial sums.By solving this, you will be able to determine the behavior of the series as $n$ approaches infinity and conclude if the series converges or diverges.

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Answered: 2 6. For the telescoping series find a…

Calculus: Early Transcendentals

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Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

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Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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2 6. For the telescoping series find a formula for the nth term of the sequence of (2k + 3)(2k + 5)' k=1 partial sums {Sn}. Then, evaluate lim Sn to obtain the value of the series or state that the series diverges. 81x