Calculus Examples of Geometric Series and More

If $a_{1}, \dots, a_{n}$ are in arithmetic progression and $a_{i} > 0$ for all i, then how to prove the following two identities:
1) $\frac{1}{\sqrt{a_{1}} + \sqrt{a_{2}}} + \frac{1}{\sqrt{a_{2}} + \sqrt{a_{3}}} + \dots + \frac{1}{\sqrt{a_{n - 1}} + \sqrt{a_{n}}} = \frac{n - 1}{\sqrt{a_{1}} + \sqrt{a_{n}}}$
2) $\frac{1}{a_{1} \cdot a_{n}} + \frac{1}{a_{2} \cdot a_{n - 1}} + \frac{1}{a_{3} \cdot a_{n - 2}} + \dots + \frac{1}{a_{n} \cdot a_{1}} = \frac{2}{a_{1} + a_{n}} (\frac{1}{a_{1}} + \frac{1}{a_{2}} + \dots + \frac{1}{a_{n}})$

caldaridvq 2022-04-20

How to evaluate $\sum_{n = 1}^{n = \infty} (\sum_{k = n}^{k = n^{2}} \frac{1}{k^{2}})$

Willie Kelley 2022-04-19

What is
$\sum_{n = 0}^{\infty} \frac{2 n^{7} + n^{6} + n^{5} + 2 n^{2}}{n!}$

Brooks Barker 2022-04-18

How do I prove this identity?
$\sum_{n = 1}^{\infty} \frac{1}{n^{s}} \sum_{n = 1}^{\infty} \frac{μ (n)}{n^{s}} = 1$

Ean Hughes 2022-04-16

How do we go from
$\frac{1}{1 + (\frac{x}{2}) + (\frac{x^{2}}{3}) + (\frac{x^{3}}{4}) + \dots}$

Alivia Rojas 2022-04-14

Is there any way to solve it?
$min {2, \frac{n}{2}} + min {3, \frac{n}{2}} + min {4, \frac{n}{2}} + \dots + min {n + 1, \frac{n}{2}} = \sum_{i = 1}^{n} min (i + 1, \frac{n}{2})$

Bradley Barron 2022-04-13

Is there a way, using the definition,
$ζ (s) = \sum_{i = 1}^{\infty} i^{- s}$
to find this?

Beckham Short 2022-04-13

How do I prove that $(v_{n})$ , defined by
$v_{n} = \frac{1}{\sqrt{n^{2} + 1}} + \frac{1}{\sqrt{n^{2} + 2}} + \dots + \frac{1}{\sqrt{n^{2} + n}},$
is convergent?

Jace Wright 2022-04-12

Radius of convergence of
$\sum_{n = 2}^{\infty} \frac{n^{2 n}}{4^{n} (2 n + 1)!} (3 - 2 x)^{n}$

Tristian Neal 2022-04-10

How do you solve this series question?
$\sum_{1}^{\infty} \frac{\cos (n π)}{\ln (6 n)}$

Papierskiix5n 2022-04-08

I want to find the Taylor series for the function:
$f (x) = \ln (\tan (x)) - \ln (x)$

Perla Benitez 2022-04-08

How to simplify $f (x) = \sum_{i = 0}^{\infty} \frac{x^{i mod (k - 1)}}{i!}$

Heather Dickson 2022-04-08

There exists a bijective $f : N \to N$ such that:
$\sum_{n = 1}^{\infty} {(- 1)}^{f (n)} \ln \frac{f (n) + 1}{f (n)} = \ln 2010$

Ekwadorkadba3 2022-04-07

How to prove
$\sum_{n = p + 1}^{\infty} \frac{1}{n^{2} - p^{2}} = \frac{1}{2 p} (1 + \frac{12}{+} \dots + \frac{1}{2 p})$

Jaylen Cantrell 2022-04-06

For each natural number n and each number $x \in [0, 1]$ let
$f_{n} (x) = \frac{x}{n x + 1}$

gabolzm6d 2022-04-06

Fix a positive number $α$ and consider the series
$\sum_{k = 1}^{\infty} \frac{1}{(k + 1) {[\ln (k + 1)]}^{α}}$

Aileen Rogers 2022-04-06

Find the value of 'x' if
$(\frac{1}{2^{\log_{x} 4}}) \cdot (\frac{1}{2^{\log_{x} 16}}) \cdot (\frac{1}{2^{\log_{x} 256}}) \dots = 2$

Breanna Mcclure 2022-04-06

How to prove that $\sum_{n = 1}^{\infty} \frac{(n - 1)!}{n \prod_{i = 1}^{n} (a + i)} = \sum_{k = 1}^{\infty} \frac{1}{{(a + k)}^{2}}$ for $a ≻ 1$ ?

palmantkf4u 2022-04-05

Find
$lim_{n \to \infty} \sum_{i = 1}^{n} \frac{2 n}{{(n + 2 i)}^{2}}$

Marzadri9lyy 2022-04-05

Evaluating
$\sum_{n = 0}^{20} \frac{{(- 1)}^{n} 2^{n + 1}}{3^{n}}$

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