Articles of sequences and series

Is it true that $\limsup \phi\le\limsup a_n$, where $\phi=\frac{a_1+…+a_n}n$?

This question already has an answer here: If $\sigma_n=\frac{s_1+s_2+\cdots+s_n}{n}$ then $\operatorname{{lim sup}}\sigma_n \leq \operatorname{lim sup} s_n$ 2 answers

Series Proof $\sum_{k=1}^n (1/k) > \ln(n+1)$

Prove that $\sum_{k=1}^n (1/k) > \ln(n+1)$. I have been trying to do this for some time now, but I cannot figure it out. It is on the study guide for my final exam, which is tomorrow so I am trying to figure it out. Thanks So I know that $\sum_{k=1}^n (1/k) = 1+1/2+1/3+1/4+1/5+\dots$, but I […]

Find $\sum_{i=, j=: i\neq j }\frac{1}{j}$

I have the following double $\sum_{i=[1,\ldots,n], j=[1,\ldots,m]: i\neq j }\frac{1}{j}$. If I ignore restriction $i \neq j$ then $$ \sum_{i=[1,\ldots,n], j=[1,\ldots,m] }\frac{1}{j}=n \sum_{ j=[1,\ldots,m] }\frac{1}{j} \le n(1+\ln(m)) $$ Now, I am not sure how to it with this restiction?

$ \sum _{n=1}^{\infty} \frac 1 {n^2} =\frac {\pi ^2}{6} $ then $ \sum _{n=1}^{\infty} \frac 1 {(2n -1)^2} $

If $ \sum _{n=1}^{\infty} \frac 1 {n^2} =\frac {\pi ^2}{6} $ then $ \sum _{n=1}^{\infty} \frac 1 {(2n -1)^2} $ Dont know what kind of series is this. Please educate. How to do such problems?

If $\sum a_n$ converges, then $\sum \sqrt{a_na_{n+1}}$ converges

Prove that if the positve term series $\sum^{\infty}_{n=1}a_n$ is convergent, also $\sum^{\infty}_{n=1}\sqrt{a_na_{n+1}}$ is convergent. Prove that if the positive term series $\sum^{\infty}_{n=1}a_n$ and $\sum^{\infty}_{n=1}b_n$ are convergent, also $\sum^{\infty}_{n=1}a_nb_n$ is convergent. I’ve tried to solve it using comparison test, but no results.

finding the limit to $(3^n+5^n)^{\frac{1}{n}}$

I was wondering if one can find the limit to the sequence $\{a_n\}$, where: $$\large a_n=(3^n+5^n)^{\large\frac{1}{n}}$$ Without the use of a calculator.

Series with cosine explanation

Would someone please explain how I would solve this problem without having to use Wolfram Alpha? This problem was originally posted on Google Plus. Wolfram Alpha:

Convergence and closed form of this infinite series?

If we have a circle of radius $r$ with an $n$-gon inscribed within this circle (i.e. with the same circumradius), we can find the difference of the areas using: $$A_n =\overbrace{\pi r^2}^\text{Area of circle}-\overbrace{\frac{1}{2} r^2 n \sin (\frac{2 \pi}{n})}^\text{Area of n-gon} =r^2(\pi-\frac{1}{2} n \sin (\frac{2 \pi}{n}))$$ I want to find the following sum (starting with […]

Looking for strictly increasing integer sequences whose gaps between consecutive elements are “pseudorandom”

I am doing some tests with strictly increasing integer sequences whose gaps between consecutive elements show a “pseudorandom” behavior, meaning “pseudorandom” that the gaps do not grow up continuously, but they change from a bigger value to a smaller one and vice versa due to the properties of the sequence without an easy way of […]

How to find the limit of the sequence $x_n =\frac{1}{2}$, if $x_0=0$ and $x_1=1$?

The formula is only applicable on values for $n\geq 2$. I know that the sequence is monotonic with a lower bound at $\frac 1 2$, but I am unsure how to find the supremum of the sequence. EDIT: $x_2 = \frac 1 2, x_3 = \frac 3 4, x_4 = \frac 5 8$. Does that […]