9.7: Sura ya 9 Mazoezi ya Mapitio
- Page ID
- 178883
Kweli au Uongo? Thibitisha jibu lako kwa ushahidi au mfano wa kukabiliana.
1) Ikiwa\(\displaystyle \lim_{n→∞}a_n=0,\) basi\(\displaystyle \sum_{n=1}^∞a_n\) hujiunga.
- Jibu
- uwongo
2) Ikiwa\(\displaystyle \lim_{n→∞}a_n≠0,\) basi\(\displaystyle \sum_{n=1}^∞a_n\) hupungua.
3) Ikiwa\(\displaystyle \sum_{n=1}^∞|a_n|\) hujiunga, kisha\(\displaystyle \sum_{n=1}^∞a_n\) hujiunga.
- Jibu
- kweli
4) Ikiwa\(\displaystyle \sum_{n=1}^∞2^na_n\) hujiunga, kisha\(\displaystyle \sum_{n=1}^∞(−2)^na_n\) hujiunga.
Je, mlolongo umepakana, monotone, na kugeuka au tofauti? Ikiwa inabadilika, pata kikomo.
5)\(a_n=\dfrac{3+n^2}{1−n}\)
- Jibu
- isiyo na mipaka, si monotone, tofauti
6)\(a_n=\ln\left(\frac{1}{n}\right)\)
7)\(a_n=\dfrac{\ln(n+1)}{\sqrt{n+1}}\)
- Jibu
- imepakana, monotone, convergent,\(0\)
8)\(a_n=\dfrac{2^{n+1}}{5^n}\)
9)\(a_n=\dfrac{\ln(\cos n)}{n}\)
- Jibu
- isiyo na mipaka, si monotone, tofauti
Je, mfululizo unaogeuka au unaojitokeza?
10)\(\displaystyle \sum_{n=1}^∞\frac{1}{n^2+5n+4}\)
11)\(\displaystyle \sum_{n=1}^∞\ln\left(\frac{n+1}{n}\right)\)
- Jibu
- hutengana
12)\(\displaystyle \sum_{n=1}^∞\frac{2^n}{n^4}\)
13)\(\displaystyle \sum_{n=1}^∞\frac{e^n}{n!}\)
- Jibu
- hukutana
14)\(\displaystyle \sum_{n=1}^∞n^{−(n+1/n)}\)
Je, mfululizo unaogeuka au unaojitokeza? Kama convergent, ni kabisa convergent?
15)\(\displaystyle \sum_{n=1}^∞\frac{(−1)^n}{\sqrt{n}}\)
- Jibu
- hujiunga, lakini si kabisa
16)\(\displaystyle \sum_{n=1}^∞\frac{(−1)^nn!}{3^n}\)
17)\(\displaystyle \sum_{n=1}^∞\frac{(−1)^nn!}{n^n}\)
- Jibu
- hukutana kabisa
18)\(\displaystyle \sum_{n=1}^∞\sin\left(\frac{nπ}{2}\right)\)
19)\(\displaystyle \sum_{n=1}^∞\cos(πn)e^{−n}\)
- Jibu
- hukutana kabisa
Tathmini.
20)\(\displaystyle \sum_{n=1}^∞\frac{2^{n+4}}{7^n}\)
21)\(\displaystyle \sum_{n=1}^∞\frac{1}{(n+1)(n+2)}\)
- Jibu
- \(\frac{1}{2}\)
22) Hadithi kutoka India inasema kwamba mtaalamu wa hisabati alinunua chess kwa mfalme. Mfalme alifurahia mchezo huo kiasi alimruhusu mwanahisabati kudai malipo yoyote. Mtaalamu wa hisabati aliomba nafaka moja ya mchele kwa mraba wa kwanza kwenye chessboard, nafaka mbili za mchele kwa mraba wa pili kwenye chessboard, na kadhalika. Kupata kujieleza halisi kwa ajili ya malipo ya jumla (katika nafaka ya mchele) ombi na mwanahisabati. Kutokana kuna\(30,000\) nafaka ya mchele katika\(1\) pauni, na\(2000\) paundi katika\(1\) tani, ngapi tani ya mchele alifanya mtaalamu wa hisabati kujaribu kupokea?
Matatizo yafuatayo yanazingatia mfano rahisi wa idadi ya watu wa nyumba, ambayo inaweza kuonyeshwa na formula ya kujirudia\(x_{n+1}=bx_n\), wapi\(x_n\) idadi ya watu wa nyumba katika kizazi\(n\), na\(b\) ni wastani wa idadi ya watoto kwa kila nyumba wanaoishi kwa kizazi kijacho. Kudhani idadi ya watu kuanzia\(x_0\).
23) Tafuta\(\displaystyle \lim_{n→∞}x_n\) kama\(b>1, \;b<1\), na\(b=1.\)
- Jibu
- \(∞, \; 0, \; x_0\)
24) Kupata kujieleza kwa\(\displaystyle S_n=\sum_{i=0}^nx_i\) katika suala la\(b\) na\(x_0\). Inawakilisha nini kimwili?
25) Kama\(b=\frac{3}{4}\) na\(x_0=100\), kupata\(S_{10}\) na\(\displaystyle \lim_{n→∞}S_n\)
- Jibu
- \(\displaystyle S_{10}≈383, \quad \lim_{n→∞}S_n=400\)
26) Kwa maana maadili\(b\) gani ya mfululizo hujiunga na kutofautiana? Mfululizo hujiunga na nini?