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Question 14
What is the sum of the following series: $$ \frac{1}{1 \times 2} + \frac{1}{2 \times 3}+\frac {1}{3 \times 4}$$ ....... $$+ \frac{1}{100 \times 101}$$?
Solution
Given series can be written as:
$$\sum_{n=1}^{100} (\frac{1}{n\times (n+1)})$$
or $$\sum_{n=1}^{100} (\frac{(n+1)-n}{n\times (n+1)})$$
or $$\sum_{n=1}^{100} (\frac{1}{n} - \frac{1}{n+1})$$
After putting values of n from 1 to 100, all terms will cancel out, only first and last terms will be there
i.e. $$1-\frac{1}{101}$$
or $$\frac{100}{101}$$
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