Calculus Examples of Geometric Series and More

How do I prove that:
$lim_{n \to \infty} \sqrt[n]{4^{n} + 9 n^{2}} = 4$

Ezekiel Yoder 2022-06-06

Compute
$\sum_{k = 2}^{\infty} \frac{(- 1)^{k} \ln k}{k}$

Summer Bradford 2022-06-06

I have the series $\sum_{n = 1}^{\infty} \frac{1 - \cos (\frac{x}{\sqrt{n}})}{\sqrt{n}}$ and I need to show that it converges pointwise for every $x \in R$

vrotterigzl 2022-06-06

Prove that $(\sum_{j = 0}^{N} a_{j}) (\sum_{j = 0}^{N} b_{j}) = \sum_{j = 0}^{2 N} c_{j}$

Felix Moore 2022-06-04

I want to calculate ${Res}_{2} z^{2} e^{\frac{1}{z - 2}}$

Mary Ashley 2022-06-03

Prove that
the following sequence is increasing
$e_{n} = {(1 + \frac{1}{n})}^{n}, n \geq 1;$

dalo2m1ezyxlo 2022-06-02

Prove that $lim_{n \to \infty} \sum_{m = 1}^{\infty} f (m, n) \geq \sum_{m = 1}^{\infty} (lim_{n \to \infty} f (m, n))$

June Salas 2022-06-01

Consider the infinite product
$P (x, p) = \prod_{n = 1}^{\infty} (1 - \frac{x}{n^{p}})$
where $x \geq 0$ and $p \geq 0$

enthral4kvri 2022-06-01

Proof verification $\sum_{n = 1}^{\infty} (- 1)^{n} \frac{1}{n}$ converges

Sage Garrison 2022-06-01

Trying to show $\sum_{n = 1}^{\infty} (\frac{1}{2 n - 1} - \frac{1}{2 \cdot 3^{n - 1}} - \frac{1}{2^{n + 1}}) = \frac{1}{2} \log 2$

virpsojolh8ih 2022-06-01

I would like study the convergence of the following sequence:
$u_{n} = \sqrt[n]{\frac{(a + 1) (a + 2) . . . (a + n)}{n!}}$

patzeriap0 2022-05-29

I would like to find an equivalent of the sequence $u_{n}$ where
$u_{n} = \int_{0}^{π / 2} \cos {(\frac{π}{2} \sin (x))}^{n} d x$

Hugo Brady 2022-05-29

Prove that
$\sum_{j = - N}^{N} e^{i \cdot j \cdot ξ \cdot λ} = \frac{\sin (N + \frac{1}{2}) \cdot (ξ \cdot λ)}{\sin (\frac{ξ \cdot λ}{2})}$

Brennen Fisher 2022-05-29

Determine if $\sum_{n = 1}^{\infty} \cos (\frac{1}{n^{2}})$ converges using the p-series approach.

meindwrhc 2022-05-29

I would like to prove convergence of the p-harmonic series
$1 + \frac{1}{2^{p}} + \frac{1}{3^{p}} + \frac{1}{4^{p}} + \dots$

Aditya Erickson 2022-05-29

Analyze the convergence behavior of the following series:
$\sum_{k = 0}^{\infty} \frac{x^{2 k}}{2^{2 k}} - \frac{x^{2 k + 1}}{3^{2 k + 1}} .$

groupweird40 2022-05-29

Prove $\sum_{n = 0}^{\infty} \frac{1}{\frac{x_{n}}{a^{n}} - 1} < \infty$ for $0 \leq a < 1 and 1 < x_{n + 1} \leq x_{n} \forall n$

Emely Baldwin 2022-05-28

I try to solve the following tricky limit:
$lim_{x \to \infty} \sum_{k = 1}^{\infty} \frac{k x}{(k^{2} + x)^{2}}$

hushjelpw4 2022-05-27

Absolute and uniform convergence of $\sum_{n = 1}^{n = \infty} 2^{n} \sin \frac{1}{3^{n} z}$

reryfaikear 2022-05-26

Let $λ > 0$ be fixed, and $a, b > 0$ positive real numbers. We have a series which is defined as
$\sum_{n = 0}^{\infty} (\prod_{j = 1}^{n} \frac{1}{1 + λ \frac{1}{a + b (j - 1)}}) .$

The majority of questions about series calculus that you will find online will have odd solutions that do not really provide explanations or the answers that would help one understand how things work. We have taken a different approach by offering series calculus practice questions that are connected with initial equations that must be solved. It is once again about estimation and analysis as the examples include the building of the bridges and calculation of the power or energy necessary to implement the safest solution by estimating the best one. By offering clear examples, we help you learn things easier!

Calculus 2

Calculus Series: Understanding Geometric Examples