☰ matematika1

Zadaci za vježbu NIZOVI I REDOVI

Rješenja

1.

, $n_{\varepsilon}=20000$ .

2.

Gomilišta niza su 0 i 3.

3.

Gomilišta niza su -1 i 1, $\inf a_n=-2$ , $\sup a_n=\displaystyle\frac{3}{2}$ , $\liminf a_n=-1$ , $\limsup a_n=1$ .

4.

$\displaystyle a=\frac{1}{3}$ , $b=-\frac{8}{3}$ .

5.

a): ,
b): $+\infty$ ,
c): ,
d): ,
e): $\displaystyle \frac{1}{\sqrt{e}}$ .

6.

a): $\displaystyle -\frac{1}{2}$ ,
b): .

7.

a): 0 ,
b): 0 .

8.

a): $\displaystyle\frac{1}{2}$ ,
b): $\displaystyle\frac{1}{3}$ ,
c): $\displaystyle\frac{1}{6}$ .

9.

$\displaystyle\frac{2}{3}$ .

10.

$-\ln{2}$ .

11.

12.

$\displaystyle\frac{1}{6}$ .

13.

$\displaystyle\frac{11}{18}$ .

14.

$\displaystyle\frac{1}{4}$ .

15.

$\displaystyle\frac{1}{3}$ .

16.

a): $\displaystyle a_n>\frac{1}{n} \textrm{ i } \sum_{n=1}^{\infty}\frac{1}{n}\textrm{ divergira} \Rightarrow\textrm{ red divergira}$ ,
b): $\displaystyle a_n<\frac{1}{n^2} \textrm{ i } \sum_{n=1}^{\infty}\frac{1}{n^2}\textrm{ konvergira} \Rightarrow\textrm{ red konvergira}$ .

17.

a): $\displaystyle \lim_{n\to\infty}\frac{a_{n+1}}{a_n}=+\infty \Rightarrow\textrm{ red divergira}$ ,
b): $\displaystyle \lim_{n\to\infty}\frac{a_{n+1}}{a_n}=\displaystyle\frac{1}{2} \Rightarrow\textrm{ red konvergira}$ .

18.

a): $\displaystyle \lim_{n\to\infty}\sqrt[n]{a_n}=\displaystyle\frac{3}{\pi} \Rightarrow\textrm{ red konvergira}$ ,
b): $\displaystyle \lim_{n\to\infty}\sqrt[n]{a_n}=+\infty \Rightarrow\textrm{ red divergira}$ .

19.

a): $\displaystyle \lim_{n\to\infty}\frac{a_{n+1}}{a_n}=0\Rightarrow\textrm{ red konvergira}$ ,
b): $\displaystyle \lim_{n\to\infty}\frac{a_{n+1}}{a_n}=+\infty \Rightarrow\textrm{ red divergira}$ ,
c): $\displaystyle \lim_{n\to\infty}\frac{a_{n+1}}{a_n}=\frac{2}{e^2}\Rightarrow\textrm{ red konvergira}$ ,
d): $\displaystyle \lim_{n\to\infty}\frac{a_{n+1}}{a_n}=0\Rightarrow\textrm{ red konvergira}$ ,
e): $\displaystyle \lim_{n\to\infty}\sqrt[n]{a_n}=0 \Rightarrow\textrm{ red konvergira}$ ,
f): $\displaystyle \lim_{n\to\infty}\sqrt[n]{a_n}=\frac{1}{e} \Rightarrow\textrm{ red konvergira}$ ,
g): $\displaystyle \lim_{n\to\infty}\sqrt[n]{a_n}=e^2 \Rightarrow\textrm{ red divergira}$ ,
h): $\displaystyle \lim_{n\to\infty}\sqrt[n]{a_n}=\frac{1}{2}\Rightarrow\textrm{ red konvergira}$ .

20.

$\displaystyle \lim_{n\to\infty}\sqrt[n]{a_n}=\frac{1}{a}\Rightarrow\textrm{red konvergira za } a>1, \textrm{divergira za } a<1$ .

Za je $\displaystyle\lim_{n\to\infty}a_n=+\infty$ pa red divergira.

21.

a): $\displaystyle \lim_{n\to\infty} n\left(1-\frac{a_{n+1}}{a_n}\right)=-\frac{1}{2}\Rightarrow\textrm{ red divergira}$ ,
b): $\displaystyle \lim_{n\to\infty} n\left(1-\frac{a_{n+1}}{a_n}\right)=\frac{3}{2}\Rightarrow\textrm{ red konvergira}$ .

22.

$\displaystyle \lim_{n\to\infty} n\left(1-\frac{a_{n+1}}{a_n}\right)=a\Rightarrow\textrm{red konvergira za } a>1, \textrm{divergira za } a<1$ .

Za je $\displaystyle a_n=\frac{1}{n+1}$ pa red divergira.

23.

a): $\displaystyle \sum_{n=0}^{\infty}\vert a_n\vert=\sum_{n=0}^{\infty}\left(\frac{1}{2}\right)^n$ je geometrijski red za $q=\frac{1}{2}<1 \Rightarrow$ red apsolutno konvergira $\Rightarrow$ red konvergira,
b): $\displaystyle \vert a_n\vert\leq\frac{1}{n^2}\,$ i $\sum_{n =1}^{\infty}\frac{1}{n^2}$ konvergira $\Rightarrow$ red apsolutno konvergira $\Rightarrow$ red konvergira.

24.

Prema Leibnizovom kriteriju redovi konvergiraju.

25.

a): $\mathbb{R}$ ,
b): ,
c): $\langle 1-e,1+e\rangle$ ,
d): $[-1,1\rangle$ ,
e): $[-10,10\rangle$ ,
f): $\langle -\infty,2\rangle\cup[8,+\infty\rangle$ ,
g): $\langle -1,0\rangle\cup\langle3,4\rangle$ .

26.

$\langle 1-e,1+e\rangle$ .

27.

a): $\displaystyle \frac{2}{1+2x}=\sum\limits_{n=0}^{\infty}(-1)^{n}2^{n+1}x^n$ , $x\in \left\langle-\frac{1}{2},\frac{1}{2}\right\rangle$ ,
b): $\displaystyle \frac{x^2}{1-x}=\sum\limits_{n=0}^{\infty}x^{n+2}$ , $x\in\langle -1,1\rangle$ ,
c): $\displaystyle \frac{x^2+1}{2+x}=\frac{1}{2}-\frac{1}{4}x+ 5 \sum \limits_{n=2}^{\infty}\left(-1\right)^n\frac{x^n}{2^{n+1}}$ , $x\in \langle-2,2\rangle$ ,
d): $\displaystyle \frac{7}{(3x+2)(2x-1)}=\sum\limits_{n=0}^{\infty}\left[\left(-\frac{3}{2}\right)^{n+1}-2^{n+1}\right]x^n$ , $x\in \left\langle-\frac{1}{2},\frac{1}{2}\right]$

28.

a): $\displaystyle \frac{1}{3-2x}=\sum\limits_{n=0}^{\infty}2^{n}(x-1)^n$ , $x\in\left\langle\frac{1}{2},\frac{3}{2}\right\rangle$ ,
b): $\displaystyle e^{2x-6}=\sum\limits_{n=0}^{\infty}\frac{2^n(x-3)^n}{n!}$ , $x\in \mathbb{R}$ ,
c): $\displaystyle \ln\sqrt{x-1}=\frac{1}{2}\sum\limits_{n=1}^{\infty}(-1)^{n+1}\frac{(x-2)^n}{n}$ , $x\in\langle 1,3]$ ,
d): $\displaystyle \ln\frac{2+x}{3+2x}=\sum\limits_{n=1}^{\infty}(-1)^{n+1}(1-2^n)\frac{(x+1)^n}{n}$ , $x\in\left\langle-\frac{3}{2},-\frac{1}{2}\right]$ ,
e): $\displaystyle \sin\left(2x-\frac{\pi}{2}\right)=\sum\limits_{n=0}^{\infty}(-1)^n\frac{2^{2n}}{(2n)!}\left(x-\frac{\pi}{2}\right)^{2n}$ , $x\in \mathbb{R}$ .

29.

$\displaystyle \sum_{n=0}^{\infty}\frac{1}{2^n\cdot n!}=\sqrt{e}$ .

30.

$\displaystyle \ln\frac{1}{\left(\frac{1}{3}x+2\right)^4}=4\ln\frac{3}{5}+4\sum\limits_{n=1}^{\infty}\frac{(-1)^{n}}{n\,5^n}(x+1)^n$ , $x\in \langle-6,4]$ , $\displaystyle \sum_{n=1}^\infty \frac{(-1)^{n-1}}{n}=\ln2$ .

Zadaci za vježbu NIZOVI I REDOVI