Intereting Posts

Norm $L^2$ of a multivariate function
What is $(-1)^{\frac{2}{3}}$?
A topology on the set of lines?
Proving $2^{2n}-1$ is divisible by $3$ for $n\ge 1$
Solve the Integral $\int \frac{dx}{\left(x-2\right)^3\sqrt{3x^2-8x+5}}$
Dirac Delta function
`“Variation of Constant”` -method to solve linear DYs?
Puzzle with pirates
Can every even integer be expressed as the difference of two primes?
Certain Sums of Conjugacy Class Sizes of Symmetric Groups
Indicator function for a vertex-induced random subgraph of $G$?
Teaching irrational numbers?
How to reparametrize curves in terms of arc length when arc-length evaluation cannot be computed analytically
Which is the easiest way to evaluate $\int \limits_{0}^{\pi/2} (\sqrt{\tan x} +\sqrt{\cot x})$?
Epsilon Induction implies Axiom Of Foundation (Or Regularity)

I know the taylor series of $\log(1+x)$. However I don’t understand how to find the convergence for $x>1$ and divergence if $x<1$.

- How prove this $\displaystyle\lim_{n\to \infty}\frac{n}{\ln{(\ln{n}})}\left(1-a_{n}-\frac{n}{\ln{n}}\right)=-1$
- Find recursive formula for sequence $a_n = \left(\frac23\right)^n + n$
- Convergence of the series $\sum_ {n\geq1} \frac {(f(n) +P(n)) \pmod {Q(n)}} {D(n)}$
- What is the fastest/most efficient algorithm for estimating Euler's Constant $\gamma$?
- Cover $\{1,2,…,100\}$ with minimum number of geometric progressions?
- Counter example to theorem in complex domain
- Convergence of series implies convergence of Cesaro Mean.
- If the positive series $\sum a_n$ diverges and $s_n=\sum\limits_{k\leqslant n}a_k$ then $\sum \frac{a_n}{s_n}$ diverges as well
- Ways to prove Eulers formula for $\zeta(2n)$
- Why does this series $\sum_{n=0}^{\infty} \frac{(n!)^{2}}{(2n)!}$ converge?

We have (i) convergence if $|x|\lt 1$, and divergence if $|x|\gt 1$. This can be done by using the Ratio Test.

We also have (ii) convergence at $x=1$ and divergence at $x=-1$. For $x=1$, we have an alternating series. For $x=-1$, we get a close relative of a familiar divergent series.

- How to solve cubic equations with given coefficients?
- Decimal/hex palindromes: why multiples of 53?
- What is a cardinal basis spline?
- Can we construct a $\mathbb Q$-basis for the Pythagorean closure of $\mathbb Q?$
- Expected number of coin tosses to land n heads
- OR-port $A\cdot B$: its conditional probabilities with zero working probability for each component? Reductio ad absurdum?
- Basic problem on topology $( James Dugundji)$
- Fourier transform of the Heaviside function
- $f(x) = |\cos x|$ prove that f is differentiable at these points and not differentiable at all other points.
- How can I determine the number of wedge products of $1$-forms needed to express a $k$-form as a sum of such?
- How to prove this equality…
- Characteristic function of Cantor set is Riemann integrable
- When can we achieve given distances between four points?
- How to prove $f(\bigcap_{\alpha \in A}U_{\alpha}) \subseteq \bigcap_{\alpha \in A}f(U_{\alpha})$?
- What do we mean by an “Elegant Proof”?