Natrag: Zadaci za vježbu Gore: NIZOVI I REDOVI  

Rješenja

1. $a=1$, $n_{\varepsilon}=20000$.

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

5. (a)

   $3$,

   (b)

   $+\infty$,

   (c)

   $e^2$,

   (d)

   $e$,

   (e)

   $$\frac{1}{\sqrt{e}}$$.

6. (a)

   $$-\frac{1}{2}$$,

   (b)

   $2$.

7. (a)

   0,

   (b)

   0.

8. (a)

   $$\frac{1}{2}$$,

   (b)

   $$\frac{1}{3}$$,

   (c)

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

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

10. $-\ln{2}$.

11. $c=6$.

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

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

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

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

16. (a)

    $$a_n>\frac{1}{n} \textrm{ i } \sum_{n=1}^{\infty}\frac{1}{n}\textrm{ divergira} \Rightarrow\textrm{ red divergira}$$,

    (b)

    $$a_n<\frac{1}{n^2} \textrm{ i } \sum_{n=1}^{\infty}\frac{1}{n^2}\textrm{ konvergira} \Rightarrow\textrm{ red konvergira}$$.

17. (a)

    $$\lim_{n\to\infty}\frac{a_{n+1}}{a_n}=+\infty \Rightarrow\textrm{ red divergira}$$,

    (b)

    $$\lim_{n\to\infty}\frac{a_{n+1}}{a_n}=\displaystyle\frac{1}{2} \Rightarrow\textrm{ red konvergira}$$.

18. (a)

    $$\lim_{n\to\infty}\sqrt[n]{a_n}=\displaystyle\frac{3}{\pi} \Rightarrow\textrm{ red konvergira}$$,

    (b)

    $$\lim_{n\to\infty}\sqrt[n]{a_n}=+\infty \Rightarrow\textrm{ red divergira}$$.

19. (a)

    $$\lim_{n\to\infty}\frac{a_{n+1}}{a_n}=0\Rightarrow\textrm{ red konvergira}$$,

    (b)

    $$\lim_{n\to\infty}\frac{a_{n+1}}{a_n}=+\infty \Rightarrow\textrm{ red divergira}$$,

    (c)

    $$\lim_{n\to\infty}\frac{a_{n+1}}{a_n}=\frac{2}{e^2}\Rightarrow\textrm{ red konvergira}$$,

    (d)

    $$\lim_{n\to\infty}\frac{a_{n+1}}{a_n}=0\Rightarrow\textrm{ red konvergira}$$,

    (e)

    $$\lim_{n\to\infty}\sqrt[n]{a_n}=0 \Rightarrow\textrm{ red konvergira}$$,

    (f)

    $$\lim_{n\to\infty}\sqrt[n]{a_n}=\frac{1}{e} \Rightarrow\textrm{ red konvergira}$$,

    (g)

    $$\lim_{n\to\infty}\sqrt[n]{a_n}=e^2 \Rightarrow\textrm{ red divergira}$$,

    (h)

    $$\lim_{n\to\infty}\sqrt[n]{a_n}=\frac{1}{2}\Rightarrow\textrm{ red konvergira}$$.

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

    Za $a=1$ je $$\lim_{n\to\infty}a_n=+\infty$$ pa red divergira.

21. (a)

    $$\lim_{n\to\infty} n\left(1-\frac{a_{n+1}}{a_n}\right)=-\frac{1}{2}\Rightarrow\textrm{ red divergira}$$,

    (b)

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

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

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

23. (a)

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

    (b)

    $$\vert a_n\vert\leq\frac{1}{n^2}$$ 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)

    $[-2,0]$,

    (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)

    $$\frac{2}{1+2x}=\sum\limits_{n=0}^{\infty}(-1)^{n}2^{n+1}x^n$$, $x\in \left\langle-\frac{1}{2},\frac{1}{2}\right\rangle$,

    (b)

    $$\frac{x^2}{1-x}=\sum\limits_{n=0}^{\infty}x^{n+2}$$, $x\in\langle -1,1\rangle$,

    (c)

    $$\frac{x^2+1}{2+x}=\frac{1}{2}-\frac{1}{4}x+ 5 \sum \limits_{n=2}^{\infty}\left(-1\right)^n\frac{x^n}{2^{n+1}}$$, $x\in \langle-2,2\rangle$,

    (d)

    $$\frac{7}{(3x+2)(2x-1)}=\sum\limits_{n=0}^{\infty}\left[\left(-\frac{3}{2}\right)^{n+1}-2^{n+1}\right]x^n$$, $x\in \left\langle-\frac{1}{2},\frac{1}{2}\right]$

28. (a)

    $$\frac{1}{3-2x}=\sum\limits_{n=0}^{\infty}2^{n}(x-1)^n$$, $x\in\left\langle\frac{1}{2},\frac{3}{2}\right\rangle$,

    (b)

    $$e^{2x-6}=\sum\limits_{n=0}^{\infty}\frac{2^n(x-3)^n}{n!}$$, $x\in\mathbb{R}$,

    (c)

    $$\ln\sqrt{x-1}=\frac{1}{2}\sum\limits_{n=1}^{\infty}(-1)^{n+1}\frac{(x-2)^n}{n}$$, $x\in\langle 1,3]$,

    (d)

    $$\ln\frac{2+x}{3+2x}=\sum\limits_{n=1}^{\infty}(-1)^{n+1}(1-2^n)\frac{(x+1)^n}{n}$$, $x\in\left\langle-\frac{3}{2},-\frac{1}{2}\right]$,

    (e)

    $$\sin\left(2x-\frac{\pi}{2}\right)=\sum\limits_{n=0}^{\infty}(-1)^n\frac{2^{2n}}{(2n)!}\left(x-\frac{\pi}{2}\right)^{2n}$$, $x\in\mathbb{R}$.

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

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

Natrag: Zadaci za vježbu Gore: NIZOVI I REDOVI