---
title: "Szeregi liczbowe"
lecture_number: 14
has_solutions: true
has_exercises: true
file_format: mystnb
kernelspec:
  name: sagemath
  display_name: SageMath
---

```{include} ../_includes/sagecell-loader.md
```

# Szeregi liczbowe

```{raw} html
<div class="sagecell-widget sagecell-exercise">
<script type="text/x-sage">

</script>
</div>
```

## Zadania na zajęcia

::::{exercise}
:label: prob-13-ex-01
:class: [exercise]
Oblicz sumę szeregu:

:::{container} columns-2-auto

1. $\displaystyle\Series \displaystyle\frac{1}{(2n-1)(2n+1)} $
2. $\displaystyle\Series \displaystyle \frac{n-1}{n!} .$
:::

:::{note}
:class: [see-lecture]

Warto spojrzeć na {numref}`lec-12-ex-02`.
:::

::::

::::{exercise}
:label: prob-13-ex-02
:class: [exercise]

Zbadaj zbieżność szeregu:

:::{container} columns-2-auto

1. $\displaystyle \Series \left[\arccos\left(\frac{n}{2n+1}\right)-\frac{\pi}{4}\right] $
2. $\displaystyle \sum_{n=3}^{\infty} \displaystyle\frac{\ln n}{n}  $
3. $\displaystyle \Series  \sin^2 \frac{1}{n} $
4. $\displaystyle \Series  \displaystyle\frac{\arcsin\left(\frac{n^2}{n^3+1}\right)}{3^n}$
5. $\displaystyle \Series  \displaystyle\frac{(n+1)!\cdot  n^n}{(2n)!\cdot  2^n}$
6. $\displaystyle \Series \displaystyle\frac{3^n\cdot n^{n^2}}{(n+1)^{n^2}} $
7. $\displaystyle \sum_{n=2}^{\infty}  \displaystyle\frac{\ln n}{n^2}.$
:::

:::{note}
:class: [see-lecture]

Warto spojrzeć na {numref}`lec-12-ex-04`, {numref}`lec-12-ex-06`, {numref}`lec-12-ex-08`, {numref}`lec-12-ex-09`.
:::

::::

::::{exercise}
:label: prob-13-ex-03
:class: [exercise]

Zbadaj zbieżność i zbieżność bezwzględną szeregu:

:::{container} columns-2-auto

1. $\displaystyle \Series (-1)^{n+1}\sin \displaystyle\frac{1}{n} $
2. $\displaystyle \Series \frac{2\sin\Paren{\sqrt n +n^2}}{\sqrt n +n^2} $
3. $\displaystyle \Series (-1)^{n+1}\left(\displaystyle\frac{2n+1}{3n+1}\right)^n .$
:::

:::{note}
:class: [see-lecture]

Warto spojrzeć na {numref}`lec-12-ex-10`, {numref}`lec-12-ex-06`, {numref}`lec-12-ex-11`.
:::

::::

## Zadania domowe

::::{exercise}
:label: prob-13-ex-04
:class: [exercise]

Zbadaj zbieżność szeregu:

:::{container} columns-2-auto

1. $\displaystyle \Series  \displaystyle\frac{n+1}{n^2+4}$
2. $\displaystyle \Series  \displaystyle\arctan\frac{n+1}{n-5}$
3. $\displaystyle \Series  \displaystyle\frac{n+1}{(n+2)\cdot 3^n}$
4. $\displaystyle \Series  \displaystyle\frac{4^n}{(4n-1)\cdot 3^n}$
5. $\displaystyle \Series  \displaystyle\frac{\Paren{n+2}^3\cdot n!}{n^n}$
6. $\displaystyle \Series  \displaystyle\frac{4^n \cdot n^{n^2}}{(n+1)^{n^2}}$
7. $\displaystyle \Series  \displaystyle\frac{(n+1)^{n^2}}{3^n \cdot n^{n^2}}$
8. $\displaystyle \Series  \displaystyle\frac{n!\cdot (n+1)^n}{\Paren{2n}!\cdot 3^n}$
9. $\displaystyle \Series  \displaystyle\frac{2^n+3^n}{3^n+4^n}.$
:::

:::{note}
:class: [see-lecture]

Warto spojrzeć na {numref}`lec-12-ex-04`, {numref}`lec-12-ex-06`, {numref}`lec-12-ex-08`, {numref}`lec-12-ex-09`.
:::

::::

::::{exercise}
:label: prob-13-ex-05
:class: [exercise]

Zbadaj zbieżność i zbieżność bezwzględną szeregu:

:::{container} columns-2-auto

1. $\displaystyle \Series (-1)^{n+1}\cos \displaystyle\frac{1}{n} $
2. $\displaystyle \Series \frac{\Paren{-5}^n}{5^n + 3^n} $
3. $\displaystyle \Series (-1)^{n+1}\left(\displaystyle\frac{n+1}{4n+1}\right)^n .$
:::

:::{note}
:class: [see-lecture]

Warto spojrzeć na {numref}`lec-12-ex-04`, {numref}`lec-12-ex-09`, {numref}`lec-12-ex-10`, {numref}`lec-12-ex-11`.
:::

::::