Can you find its general term?
The value of the summation above can be expressed as where and are coprime positive integers. Find the value of
The answer is 26.
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2 solutions
Moderator note:
Nice job. But it would be better to express in terms of its partial fractions first.
n = 1 ∑ 2 0 n 2 + n 2 n ( n 2 + 1 ) ⇒ a + b = 5 + 2 1 = n = 1 ∑ 2 0 n ( n + 1 ) 2 n ( n 2 + n − n + 1 ) = n = 1 ∑ 2 0 2 n − n = 1 ∑ 2 0 n + 1 2 n + n = 1 ∑ 2 0 n ( n + 1 ) 2 n = n = 1 ∑ 2 0 2 n − n = 1 ∑ 2 0 n + 1 2 n + n = 1 ∑ 2 0 n 2 n − n = 1 ∑ 2 0 n + 1 2 n = n = 1 ∑ 2 0 2 n + n = 1 ∑ 2 0 n 2 n − n = 1 ∑ 2 0 n + 1 2 n + 1 = n = 1 ∑ 2 0 2 n + n = 1 ∑ 2 0 n 2 n − n = 2 ∑ 2 1 n 2 n = 2 − 1 2 ( 2 2 0 − 1 ) + 1 2 1 − 2 1 2 2 1 = 2 2 1 − 2 + 2 − 2 1 2 2 1 = 2 1 ( 2 1 − 1 ) 2 2 1 = 2 1 2 0 ˙ 2 2 1 = 2 1 5 2 2 3 = 2 6