# Questions tagged [convergence]

Convergence of sequences and different modes of convergence.

14,571 questions
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### if $\sum_{n=1}^{\infty} a_n$ converges absolutely then prove that $\sum_{n=1}^{\infty} a_n^2$ converges [duplicate]

I am self studying real analysis.I have come up with following proof and I know that other proofs exist. But, Can someone just tell me if there is anything wrong with the following proof.Thanks in ...
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### Problem Concerning Cauchy Principle for Sequences.

I have a question but can't seem to figure out how to solve it. The problem states: Let's consider a sequence $x_n$, such that $x_n\to a$, as $n \to \infty$. Using the Cauchy Principle prove that (a)...
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### Show power series converges for every $x$.

Let $$f(x) = 1 + a_{1}x^{1}+a_{2}x^{2}+a_{3}x^{3}+a_{4}x^{4}+...$$ be a solution of the differential equation $f'(x)=xf(x).$ Now I need to explain that the power series that define $f(x)$ converges ...
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