Evaluate 1 + 3/4 + (3.5)/(4.8) + (3.5.7)/(4.8.12) +...
I have simplified it to Summation ((2k+1)!)/(k!)(k!)) (1/8)^k where k varies from 0 to infinity.
I am not able to to relate this sum with Taylor expansion of some function
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https://math.stackexchange.com/questions/746388/calculating-1-frac13-frac1-cdot33-cdot6-frac1-cdot3-cdot53-cdot6-cdot – lab bhattacharjee Jan 09 '19 at 05:17
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Sounds like you are looking for $$ \sum_{k = 0}^\infty (2k+1) \binom{2k}{k}\left( \frac18 \right)^k, $$ so consider differentiating $$ f(x) = \sum_{k = 0}^\infty \binom{2k}{k} x^{2k+1}, $$ and using $(1/8)^k = (1/\sqrt8)^{2k}$
gt6989b
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the series is the binomial expansion of $$ \bigg(1 - \frac12 \bigg)^{-\frac32} $$
which gives the sum $2\sqrt{2}$
Ross Millikan
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David Holden
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1MathJax hint: if you use \left and \right before parentheses and other delimiters they autosize based on what is inside. – Ross Millikan Jan 09 '19 at 05:43