Reputation
108,015
Next tag badge:
99/100 score
26/20 answers
Badges
8 92 164
Impact
~395k people reached

4m
comment Real analysis and taylor series
Since $a_0 = 0$, the terms $a_0\cdot a_n$ and $a_n\cdot a_0$ can be omitted. For uniformity of expression, they should be omitted, makes things easier. For the $H_n$ vs. $H_{n-1}$ thing, note that $c_n$ is the coefficient of $z^n$, your given series is written with exponents $n+1$.
24m
comment Real analysis and taylor series
Here's a hint: $n = (n-k) + k$.
40m
comment Real analysis and taylor series
Right. Now note that here $a_k = b_k$, since we're squaring, and that $a_0 = 0$, so you get $$c_n = \sum_{k=1}^{n-1} a_k\cdot a_{n-k}.$$ Then fill in what $a_k$ is. You get a sum, and you want to show that that equals $$c_n = \frac{2H_{n-1}}{n}.$$ The natural first step is to multiply both sides of the should-be-equation by $n$, then play around a bit to try and see the equality. If you find the right turn, it's quite easy. (If you don't, it's not easy.)
1h
comment Real analysis and taylor series
You lost a minus sign there, but since we're squaring it, that's not a deal-breaker. Okay, now the Cauchy product. What was the general formula for the coefficients of the Cauchy product of two power series?
1h
comment difficult limit with a improper integral
You know how $e^x$ behaves at $0$, I presume. And you probably also know the limit of $\frac{\sin x}{x}$. From that, you can determine the limit. Then replace that expression with its limit - afterwards you need to show that that is legitimate, but first, just do it - and compute the limit of the integrals of the remaining factor.
1h
comment difficult limit with a improper integral
The lower limit of the integral is $(n+1)^{-2}$, that changes things a bit.
1h
revised Real analysis and taylor series
LaTeXification for readability
1h
comment Real analysis and taylor series
What did you get for the Taylor series of $\log (1-z)$?
1h
answered Every locally finite family of non-empty subsets of a Lindelöf space is countable.
1h
revised Let $f$ be a non-constant entire function. Prove that $f(z)=cz^n$ for some constant $c$ and positive integer $n$
Cosmetics for better readability
2h
comment difficult limit with a improper integral
The first factor has a limit at $0$, so you basically need only look at the second.
2h
comment difficult limit with a improper integral
$$\frac{e^x\sin^2(x)}{x^{\frac{7}{2}}} = \frac{e^x\sin^2(x)}{x^2}\cdot \frac{1}{x^{\frac{3}{2}}}$$
4h
reviewed Leave Closed Can I use my powers for good?
4h
revised Identifying simple examples of fields
Some formatting and LaTeX
5h
reviewed Leave Closed Remarkable mathematics in the The Simpsons television show
5h
comment Countable and not closed subset of infinite compact space
Do you know that in a compact space, every infinite subset has a limit point?
5h
comment Weak and weak$^*$ topologies
Yes, that is right. If $F$ is a vector space of linear forms on the vector space $E$, then one denotes the topology generated by the inverse images of open subsets of $\mathbb{K}$ under the functionals in $F$ by $\sigma(E,F)$. Thus, the weak topology on $X$ is $\sigma(X,X^\ast)$, and the weak$^\ast$ topology on $X^\ast$ is (with the canonical identification of $X$ with a subspace of $X^{\ast\ast}$) $\sigma(X^\ast, X)$ - whereas the weak topology on $X^\ast$ is $\sigma(X^\ast, X^{\ast\ast})$. The weak and weak$^\ast$ topology on $X^\ast$ coincide if $X$ is reflexive.
5h
answered Compute an integral with Cauchy's residue theorem
6h
comment How to prove the function $f$ has an antiderivative?
$G$ isn't differentiable at $0$. For $f$, the thing to show is that $$\lim_{x\downarrow 0} \frac{1}{x}\int_0^x \sin \frac{\pi}{t}\,dt = 0.$$
6h
comment Compute an integral with Cauchy's residue theorem
Your definition of $\gamma$ seems wrong. $\gamma(t) = -3^{2i\pi t} = -e^{2i\pi t\log 3}$ doesn't give a closed curve for the parameter interval $[0,3]$. Should it have been $\gamma(t) = -3 e^{2i\pi t}$?