⧼exchistory⧽
6 exercise(s) shown, 0 hidden
BBy Bot
Nov 03'24
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[/math]
Classify each of the following infinite series as absolutely convergent, conditionally convergent, or divergent. Show how you obtain your answer starting from a standard test or series.
- [a [math]\sum_{i=0}^\infty (-1)^i \frac{1}{2i-3}[/math]
- [math]\sum_{k=1}^\infty \frac{1}{(k^3+1)^{\frac12}}[/math]
- [math]\sum_{k=1}^\infty (-1)^k \frac{1}{(k+1)^{\frac23}}[/math]
- [math]\sum_{i=1}^\infty \frac{i2^i}{3^{i+1}}[/math]
- [math]\sum_{n=1}^\infty (-1)^n \frac{5^n}{4^{n+1}}[/math]
- [math]\sum_{k=0}^\infty \frac{100^k}{k!}[/math]
- [math]\sum_{k=1}^\infty (-1)^k \frac{k!}{100k}[/math]
- [math]\sum_{i=1}^\infty (-1)^i e^{-i^2}[/math]
BBy Bot
Nov 03'24
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[/math]
- Prove that the series [math]\sum_{n=0}^\infty \frac{n}{2^n}[/math] is absolutely convergent.
- Prove that [math]\lim_{n\goesto\infty} \frac{n}{2^n} = 0[/math].
BBy Bot
Nov 03'24
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[/math]
Classify each of the following series as absolutely convergent, conditionally convergent, or divergent. Show how you obtain your answer.
- [math]\sum_{n=1}^\infty \frac{\ln n}n[/math]
- [math]\sum_{n=1}^\infty (-1)^n \frac{\ln n}n[/math]
- [math]\sum_{n=1}^\infty (-1)^n \frac{\ln n}{n^3}[/math]
- [math]\sum_{n=1}^\infty \frac{\ln n}{n^2}[/math].
BBy Bot
Nov 03'24
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[/math]
- Prove that, for every positive number [math]a[/math], the series [math]\sum_{i=0}^\infty \frac{a^i}{i!}[/math] is absolutely convergent.
- Prove that [math]\lim_{n\goesto\infty} \frac{a^n}{n!} = 0[/math] for every positive number [math]a[/math].
BBy Bot
Nov 03'24
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[/math]
The infinite series
[[math]]
\sum_{n=0}^\infty a_n = 1 + \frac12 + \frac1{2\cdot3} +
\frac1{2^23} + \frac1{2^23^2} + \frac1{2^33^2} +
\frac1{2^33^3} + \cdots
[[/math]]
is defined, for every integer [math]n\geq0[/math], by the two equations:
[[math]]
a_{2n} = \frac1{2^n3^n}
[[/math]]
[[math]]
a_{2n+1} = \frac1{2^{n+1}3^n}
.
[[/math]]
- Show that [math]\sum_{n=0}^\infty a_n[/math] is absolutely convergent.
- What is [math]\lim_{n\goesto\infty} \frac{|a_{n+1}|}{|a_n|}[/math]?
BBy Bot
Nov 03'24
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[/math]
As a corollary of \ref{thm 9.5.1}, prove the following extension of the Comparison Test: The series [math]\sum_{i=m''^\infty a_i[/math] is absolutely convergent if there exists an absolutely convergent series [math]\sum_{i=m}^\infty b_i[/math] such that [math]|a_i| \leq |b_i|[/math] eventually.}