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Expression of the sum of a power series

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#sequences-and-series #discrete-mathematics #power-series #arithmetic-progressions
531
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How does this infinite series $1-\frac{1}{4}+\frac{1}{7}-\frac{1}{10}+\cdots$ simplify to an integral $\int_0^1\frac{dx}{1+x^3}$?

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#integration #sequences-and-series #power-series
45
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What is the radius of convergence for $\sum_{j=0}^{\infty}\frac{i^{j}+(-i)^{j}-i}{j!}z^{j}?$

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#sequences-and-series #complex-analysis #convergence-divergence #power-series
97
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Find the Taylor series of $\frac{1+z}{1-z}$ at $z_{0}=i$

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#sequences-and-series #summation #power-series #taylor-expansion
97
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Example III.3.2(1) of Baumslag's "Topics in Combinatorial Group Theory": proving $F=\operatorname{gp}(1+\xi\mid \xi\in\Xi)$ is free.

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#group-theory #power-series #noncommutative-algebra #free-groups #combinatorial-group-theory
1.9k
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Derive the sum of $\sum_{i=1}^n ix^{i-1}$

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#sequences-and-series #convergence-divergence #summation #power-series
179
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Topological Algebraic Independence of power series

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#number-theory #convergence-divergence #prime-numbers #power-series #p-adic-number-theory
39
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Converting a function to a power series with only variables. A bit stuck

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#sequences-and-series #power-series
519
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Find a composite solution to the following problem

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#ordinary-differential-equations #power-series #approximation
119
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Could the series $\sum (x-3)^n/n$ be seen as a power series if we consider $1/n$ as $c_n$?

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#calculus #sequences-and-series #power-series
77
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Can I find a power series representation for $\frac {1}{(1+x)^2}$ ONLY by differentiating $\frac {1}{1+x}$?

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#calculus #sequences-and-series #power-series
2k
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How do I formally show the radius of convergence of the Taylor series of $f(x)=x^6 - x^4 + 2$ at $a=-2$?

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#calculus #sequences-and-series #power-series #taylor-expansion
109
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How many solutions does equation $\int\limits_x^{x+\frac{1}{2}} \cos \left( \frac{t^2}{3} \right) dt = 0$ have on the segment [0, 3]?

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#integration #sequences-and-series #power-series #taylor-expansion #integral-equations
3.1k
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How can Taylor series diverge?

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#calculus #power-series #taylor-expansion #divergent-series
33
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Summing a general power series

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#power-series

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