SymPy Gamma

SymPy:

summation ((1/n)*x**n,(n,1,oo ))

$\sum_{n=1}^{\infty} \frac{x^{n}}{n}$

Evaluate with respect to…

Result:

$\begin{cases} - \log{\left(1 - x \right)} & \text{for}\: x \geq -1 \wedge x < 1 \\\sum_{n=1}^{\infty} \frac{x^{n}}{n} & \text{otherwise} \end{cases}$

Plot:

Roots:

solve(Piecewise((-log(1 - x), (x >= -1) & (x < 1)), (Sum(x**n/n, (n, 1, oo)), True)), x)

$x =$

Derivative:

diff(Piecewise((-log(1 - x), (x >= -1) & (x < 1)), (Sum(x**n/n, (n, 1, oo)), True)), x)

$\frac{d}{d x} \begin{cases} - \log{\left(1 - x \right)} & \text{for}\: x \geq -1 \wedge x < 1 \\\sum_{n=1}^{\infty} \frac{x^{n}}{n} & \text{otherwise} \end{cases} =$

Antiderivative forms:

Series expansion around 0:

series(Piecewise((-log(1 - x), (x >= -1) & (x < 1)), (Sum(x**n/n, (n, 1, oo)), True)), x, 0, 10)

See what Wolfram|Alpha has to say.

Experiment with SymPy at SymPy Live.