Intereting Posts

How prove this inequality $\left(\int_{0}^{1}f(x)dx\right)^2\le\frac{1}{12}\int_{0}^{1}|f'(x)|^2dx$
An exotic sequence
Is There a Problem with This Professor's Proof Concerning Interior and Boundary Points?
Largest eigenvalue of a positive semi-definite matrix is less than or equal to sum of eigenvalues of its diagonal blocks
Convex optimization with non-convex objective function
Infinite Sum without using $\sin\pi$
Munkres Topology, page 102, question 19:a
An exercise of the book “Hamilton's Ricci Flow” by Bennett Chow
Are there other cases similar to Herglotz's integral $\int_0^1\frac{\ln\left(1+t^{4+\sqrt{15}}\right)}{1+t}\ \mathrm dt$?
Integral $\int_0^{2\pi}\frac{dx}{2+\cos{x}}$
Maximizing a linear function over an ellipsoid
Uniform convergence of $\sum_{n=0}^{\infty} \frac{\int_{\sin nx}^{\sin(n+1)x}\sin t^2dt \int_{nx}^{\infty}\frac{dt}{\sqrt{t^4+1}}}{1+n^3x^2}$
What is Cauchy Schwarz in 8th grade terms?
How do I calculate generalized eigenvectors?
Lebesgue integral and a parametrized family of functions

I am investigating the convergence of

$$\begin{split}\sum _{n=1}^{\infty }\left\{ \dfrac {1\cdot 3\cdots (2n-1)} {2\cdot 4\cdots (2n)}\cdot \dfrac {4n+3} {2n+2}\right\} ^{2} &= \sum _{n=1}^{\infty }\left\{ \dfrac {\prod _{t=1}^n (2t-1)} {\prod _{t=1}^n (2t)}\cdot \dfrac {4n+3} {2n+2}\right\} ^{2} \\

&=\sum _{n=1}^{\infty }\left\{ \prod _{t=1}^n\left( 1-\dfrac {1} {2t}\right) \dfrac {4n+3} {2n+2}\right\} ^{2}

\end{split}$$

which after some manipulations I have reduced to

$$\sum _{n=1}^{\infty }e^ \left\{ 2\ln \left(2 -\dfrac {1} {2n+2}\right) +2\cdot \sum _{t=1}^{n}\ln \left( 1-\dfrac {1}{2t}\right) \right\} $$

and from an alternative approach I was able to reduce it to

$$\sum _{n=1}^{\infty } \dfrac{\left( 4n+3\right) ^{2}}{4\left(n+1\right)^{2}} \prod _{t=1}^n\left( 2+\dfrac{1}{2t^{2}}-\dfrac{2}{t}\right)$$

I am unsure how to proceed from here in either of the two cases. Any help would be much appreciated.

- Uniform continuity question
- Limits: How to evaluate $\lim\limits_{x\rightarrow \infty}\sqrt{x^{n}+a_{n-1}x^{n-1}+\cdots+a_{0}}-x$
- e as sum of an infinite series
- Does there exist a bijective $f:\mathbb{N} \to \mathbb{N}$ such that $\sum f(n)/n^2$ converges?
- Infinite limits
- Limit of nth root of n!
- Interpretation of $\epsilon$-$\delta$ limit definition
- Can we use this formula for a certain indeterminate limit $1^{+\infty}$?
- Do the sequences from the ratio and root tests converge to the same limit?
- Finding the limit $\displaystyle\lim_{x\to 0+} \left(\frac{\sin x}x\right)^{1/{x^2}}$

We can prove by induction that

$$\dfrac {1\cdot 3\cdots (2n-1)} {2\cdot 4\cdots (2n)} \ge \frac{1}{\sqrt{4n}}$$

and so your series diverges.

You can also notice that

$$\dfrac {1\cdot 3\cdots (2n-1)} {2\cdot 4\cdots (2n)} = \dfrac{\binom{2n}{n}}{4^n}$$

and try using the approximation

$$ \dfrac{\binom{2n}{n}}{4^n} = \frac{1}{\sqrt{\pi n}} \left(1 + \mathcal{O}\left(\frac{1}{n}\right)\right)$$

Denote by $a_n$ the general term, which is positive. We can rewrite it as $\left(\frac{(2n)!}{4^nn!n!}\right)^2\left(\frac{4n+3}{2n+2}\right)^2$, which is equivalent to $b_n:=4\left(\frac{(2n)!}{4^nn!n!}\right)^2$. Now we use Stirling’s formula, which states that $n!\overset{+\infty}{\sim}\left(\frac ne\right)^n\sqrt{2\pi n}$. We get

\begin{align*}

b_n&\overset{+\infty}{\sim} 4\left(\frac{\left(\frac{2n}e\right)^{2n}\sqrt{4n\pi}}{4^n\left(\frac ne\right)^{2n}2\pi n}\right)^2\\

&=\frac 4{n\pi},

\end{align*}

and using the fact that the harmonic series diverges, we get that the series $\sum_n a_n$ is divergent.

- What is the solution to the Dido isoperimetric problem when the length is longer than the half-circle?
- Finite partition of a group by left cosets of subgroups
- Reinventing The Wheel – Part 1: The Riemann Integral
- Embedding of free $R$-algebras
- Proving a function is continuous and periodic
- Why $y=e^x$ is not an algebraic curve?
- Finding the Ideals of a Direct Product of Rings
- The cone is not immersed in $\mathbb{R}^3$
- Use mathematical induction to prove that any integer $n\ge2$ is either a prime or a product of primes.
- Quasi Cauchy sequences in general topology?
- Real Numbers to Irrational Powers
- Trigonometric Triangle Equality
- Why can't erf be expressed in terms of elementary functions?
- Intruiging Symmetric harmonic sum $\sum_{n\geq 1} \frac{H^{(k)}_n}{n^k}\, = \frac{\zeta{(2k)}+\zeta^{2}(k)}{2}$
- What's the densitiy of the product of two independent Gaussian random variables?