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Differentiation to Solve Modified Sum of Binomial Coefficients

143.Summation...

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143.Summation, k=1 to 2015 ipower 1/(4m+k) where mN

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Q. Compute the summation

\sum_{k = 0}^{27} k (_{k}^{27}) (\frac{1}{2})^{k} (\frac{2}{3})^{27 - k}

A_{k} = [\begin{matrix} k & k - 1 k - 1 & k \end{matrix}]

| A_{1} | + | A_{2} | + . . + | A_{2015} | = ?

\sum_{r = 1}^{n} [\sum_{k = 1}^{r} k] {[{log}_{1 / 2} \sqrt{(4 x - x^{2})}]}_{1 / 2}^{r}

x

for which summation is a finite number as

n \to \infty

5 \sum n = 1 5 \sum m = 1 {tan}^{- 1} \frac{m}{n} = k π, k \in R,

[k] =

[.]

denotes greatest integer function)

Q. If the sum of

1 + \frac{3}{2} + \frac{5}{4} + \frac{7}{8} + \dots n

= S = k - \frac{m n + 3}{m^{n - 1}}

k - m

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Differentiation to Solve Modified Sum of Binomial Coefficients

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