Is 6 = 2 x 3? You'll be greatly messing this up!
k = 1 ∑ ∞ ( 3 k − 2 k ) ( 3 k + 1 − 2 k + 1 ) 6 k = ?
The answer is 2.
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1 solution
Hurray! Good Solutions!
@Tanishq Varshney can you please explain how did you arrive at the partial fraction decomposition of the given expression?
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You can rewrite the expression as
2 ( ( 3 / 2 ) k − 1 ) ( 2 3 ( 3 / 2 ) k − 1 ) ( 3 / 2 ) k
Hint take 2 k common from each term in denominator and let ( 3 / 2 ) k = t and then it's simple partial fraction
the series can be rewritten as
n → ∞ lim k = 1 ∑ n ( 3 k − 2 k 3 k − 3 k + 1 − 2 k + 1 3 k + 1 )
its a telescopic series, thus the expression reduces to
n → ∞ lim 3 − 3 n + 1 − 2 n + 1 3 n + 1
3 − n → ∞ lim 1 − ( 3 2 ) n + 1 1
3 − 1 = 2