(a) Show that11 2 +12 3 +13 4 ++1n(n + 1) = nn + 1Hint:1n(n + 1) = 1n 1n + 1(b) Use the

Chapter 5, Problem 62

(choose chapter or problem)

(a) Show that

$\frac{1}{1-2}+\frac{1}{2-3}+\frac{1}{3 \cdot 4}+\ldots+\frac{1}{n(n+1)}=\frac{n}{n+1}$

$\text { [Hint: } \left.\frac{1}{n(n+)}=\frac{1}{n}-\frac{1}{n+1}\right]$

(b) Use the result in part (a) to find

$\lim _{n \rightarrow+\infty} \sum_{k=1}^{n} \frac{1}{k(k+1)}$

Equation Transcription:

$\frac{1}{1\cdot{}2}+\frac{1}{2\cdot{}3}+\frac{1}{3\cdot{}4}+...+\frac{1}{n(n+1)}=\frac{n}{n+1}$

[Hint: $\frac{1}{n(n+)}=\frac{1}{n}-\frac{1}{n+1}$ ]

$\lim\limits_{{n} \rightarrow {+\infty{}}}\sum\limits_{k=1}^{n}\frac{1}{k(k+1)}$

Text Transcription:

1/ 1.2 +1/ 2.3 +1/ 3.4 +...+ 1/n(n+1) =n/n+1

[Hint: 1/n(n+1) =1/n - 1/n+1 ]

lim _n right arrow + infinity sum_k=1 ^n 1/k(k+1)

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