Summation to n of Power of k over k/Proof 2

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Theorem

$\ds \sum_{k \mathop = 1}^n \dfrac {x^k} k = H_n + \sum_{k \mathop = 1}^n \dbinom n k \dfrac {\paren {x - 1}^k} k$

where:

$H_n$ denotes the $n$th harmonic number
$\dbinom n k$ denotes a binomial coefficient.


Proof

Differentiating with respect to $x$:

\(\ds \dfrac \d {\d x} \paren {\sum_{k \mathop = 1}^n \dfrac {x^k} k}\) \(=\) \(\ds \sum_{k \mathop = 1}^n\dfrac \d {\d x}\paren {\dfrac {x^k} k}\) Applications of Linear Combination of Derivatives
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 1}^n k \dfrac {x^{k - 1} } k\) Power Rule for Derivatives
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 1}^n x^{k - 1}\)
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 0}^{n - 1} x^k\) Translation of Index Variable of Summation
\(\ds \) \(=\) \(\ds \dfrac {x^n - 1} {x - 1}\) Sum of Geometric Sequence


Thus:

\(\ds \sum_{k \mathop = 1}^n \dfrac {x^k} k\) \(=\) \(\ds \int \dfrac {x^n - 1} {x - 1} \rd x\)
\(\ds \) \(=\) \(\ds \int \dfrac {\paren {z + 1}^n - 1} z \rd z\) Integration by Substitution: $z = x - 1$
\(\ds \) \(=\) \(\ds \int \paren {\dfrac {\paren {z + 1}^n} z - \dfrac 1 z} \rd z\)
\(\ds \) \(=\) \(\ds \int \paren {\dfrac 1 z \sum_{k \mathop = 0}^n \dbinom n k z^k - \dfrac 1 z} \rd z\) Binomial Theorem
\(\ds \) \(=\) \(\ds \int \paren {\sum_{k \mathop = 1}^n \dbinom n k z^{k - 1} + \dfrac 1 z - \dfrac 1 z} \rd z\) Binomial Coefficient with Zero
\(\ds \) \(=\) \(\ds \int \paren {\sum_{k \mathop = 1}^n \dbinom n k z^{k - 1} } \rd z\)
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 1}^n \dbinom n k \int z^{k - 1} \rd z\) Linear Combination of Primitives
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 1}^n \dbinom n k \dfrac {z^k} k + C\) Primitive of Power
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 1}^n \dbinom n k \dfrac {\paren {x - 1}^k} k + C\)


When $x = 1$ we have:

\(\ds \sum_{k \mathop = 1}^n \dbinom n k \dfrac {\paren {1 - 1}^k} k + C\) \(=\) \(\ds \sum_{k \mathop = 1}^n \dfrac {1^k} k\)
\(\ds \leadsto \ \ \) \(\ds 0 + C\) \(=\) \(\ds \sum_{k \mathop = 1}^n \dfrac 1 k\)
\(\ds \leadsto \ \ \) \(\ds C\) \(=\) \(\ds H_n\) Definition of Harmonic Number

The result follows.


Sources

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