Solving Series: Techniques, Examples, and Practice Problems

If $f (n) = [\sqrt{n} + \frac{1}{2}]$ when n is a natural number and [⋅] represents the floor function, then find the value of
$\sum_{n = 1}^{\infty} \frac{2^{f (n)} + 2^{- f (n)}}{2^{n}}$

oleifere45 2022-06-22

Let $(a_{n})$ be a bounded sequence of positive real numbers such that $lim_{n \to \infty} \sqrt[n]{a_{1} \dots a_{n}} = 1.$ Prove that $lim_{n \to \infty} \frac{a_{1} + \dots + a_{n}}{n} = 1.$

Armeninilu 2022-06-22

Find sum:
$\sum_{i = k}^{n} 3$

Roland Manning 2022-06-21

Сheck сonvergence $\sum_{n = 2}^{\infty} \frac{\sqrt[n]{n}}{\ln^{2} n}$

manierato5h 2022-06-21

Convergence of a recursevely defined sequence $x_{0} = 0$ , $x_{n + 1} = 1 - (1 / 2) x_{n}^{2}$

cazinskup3 2022-06-18

I‘m trying to understand this equality:
$e^{z + z^{- 1}} = \sum_{n = - \infty}^{\infty} (\sum_{m = 0}^{\infty} \frac{1}{(m + n)! m!}) z^{n}$

enrotlavaec 2022-06-16

Suppose $\sum_{n = 0}^{\infty} a (n) < \infty$ , what about $\sum_{n = 2}^{\infty} a (\frac{n}{\log (n)})$ ?

vittorecostao1 2022-06-15

Determine whether the series
$\sum_{n = 1}^{\infty} \frac{(\frac{2}{3} + \frac{1}{3} \cdot \sin (n))^{n}}{n}$
converges.

Hector Petersen 2022-06-15

Evaluating series $\sum_{n = 1}^{\infty} n^{2} (1.05)^{- n}$

taghdh9 2022-06-14

Proving $\sum_{i = 1}^{n} i \sin i x = \frac{\sin ((n + 1) x)}{4 \sin^{2} \frac{x}{2}} - \frac{(n + 1) \cos ((2 n + 1) \frac{x}{2})}{2 \sin \frac{x}{2}}$ by induction

Sattelhofsk 2022-06-14

Prove $\frac{1}{u_{0} u_{1}} + \frac{1}{u_{1} u_{2}} + \dots + \frac{1}{u_{i} u_{i + 1}} + \dots \dots + \frac{1}{u_{n} u_{n + 1}} = \frac{n + 1}{u_{0} u_{n + 1}}$

Brunton39 2022-06-13

Limit of summation of $\sum_{i = 0}^{\infty} e^{- i k}$ series

Abram Boyd 2022-06-13

I am trying to compute the integral
$\begin{matrix} (1) & \int_{0}^{π / 3} \frac{x}{\cos (x)} d x \end{matrix}$

cazinskup3 2022-06-13

I want to expand the function following into Laurent series:
$\frac{z^{2} + i z - 1}{(z - 2) (z + 3)}$
for $2 < | z | < 3$

Santino Bautista 2022-06-11

Evaluating the sum $lim_{t \to \infty} \sum_{k = t}^{\infty} f (k)$

Leland Morrow 2022-06-11

I'm looking for a closed-form for the following sum:
$\sum_{n = 1}^{\infty} (m n arccoth (m n) - 1)$ for $| m | > 1.$

Davion Harding 2022-06-09

If sequence $a_{n} > 0$ is decreasing to 0 and $\sum_{k = 1}^{\infty} a_{n} = \infty$ , prove $\sum_{n = 1}^{\infty} \frac{a_{n}}{e^{S_{n}}} < \infty$

hicaderasu1c8q 2022-06-04

$1^{2} - 2^{2} + 3^{2} - 4^{2} + \dots + 1999^{2}$ 19992 using closed form formula for sum of squares of first n natural numbers

trojanman849089tck 2022-06-04

Calculate the sum function for the power series $\sum_{n = 0}^{\infty} \frac{1}{4 n^{2} - 1} x^{2 n + 1}$

Reginald Hanna 2022-06-03

Show that $\frac{(\binom{n}{1})}{1} + \frac{(\binom{n}{2})}{2} + \frac{(\binom{n}{3})}{3} + \dots + \frac{(\binom{n}{n})}{n} = \sum_{r = 1}^{n} \frac{2^{r} - 1}{r}$

Series math problems relate to the precalculus stage of mathematical studies that are met both by high school students and college learners dealing with analysis. The questions that are brought up by this specific approach will include solving series equations that represent the sum of a sequence to a certain number of terms. You can take a look at various series math examples that will help you approach series math questions that may relate either to statistical calculations or engineering equations that are mostly used in engineering and data programming.

Precalculus

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