Find a formula for the nth term of a sequence of partial sums. Then evaluate...

Question:

Find a formula for the nth term of a sequence of partial sums. Then evaluate {eq}\lim\limits_{n\rightarrow\infty}S_n {/eq} to find the limiting value or show that the series diverges.

1. {eq}\sum\limits_{k=1}^\infty\left(\dfrac{1}{k+1}-\dfrac{1}{k+2}\right) {/eq}

2. {eq}\sum\limits_{k=1}^\infty\dfrac{1}{(2k+1)(2k+3)} {/eq}

Telescoping Series:

Series which can be present as the partial sum such that which all term get cancel except it's first and the last term then it has finite sum this type of series is called telescoping series. Telescoping series is always convergent as it has a finite sum.

Let {eq}{k_n} {/eq} is the general term of the series then it can be reduced into; {eq}{k_n} = {m_n} - {m_{n + 1}} {/eq}

Answer and Explanation:

{eq}\displaystyle \eqalign{ & 1. \cr & \sum\limits_{k = 1}^\infty {\left( {\frac{1}{{k + 1}} - \frac{1}{{k + 2}}} \right)} \cr &...

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