Question:
Find the sum \(S_n\) and \(S_\infty\) of given series:
\(\frac{1}{1 \cdot 3} + \frac{1}{3 \cdot 5} + \frac{1}{5 \cdot 7} + \cdots + \frac{1}{(2n-1)(2n+1)}\)
Solution:
These types of series are considered as “special series”. We can definately find a pattern among terms but we can't find sum by general formulas (as we do in AP and GP). So, here we need to modify the given series and hence we will reach to the solution.
Given:
\(\frac{1}{1 \cdot 3} + \frac{1}{3 \cdot 5} + \frac{1}{5 \cdot 7} + \cdots + \frac{1}{(2n-1)(2n+1)}\)
This can be written as:
\(\frac{1}{2}(\frac{3-1}{1 \cdot 3}) + \frac{1}{2}(\frac{5-3}{3 \cdot 5}) + \frac{1}{2}(\frac{7-5}{5 \cdot 7}) + \cdots\)
\(+ \frac{1}{2}(\frac{(2n+1)-(2n-1)}{(2n-1)(2n+1)})\)
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