⧼exchistory⧽
8 exercise(s) shown, 0 hidden
BBy Bot
Nov 03'24
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[/math]
Determine whether or not each of the following infinite series converges, and evaluate it if it does.
- [a [math]\sum_{i=0}^\infty \frac7{5^i}[/math]
- [math]\sum_{k=1}^\infty \frac{a}{5^k}[/math]
- [math]\sum_{n=1}^\infty \left( \frac1{2^n} + \frac1n \right)[/math]
- [math]\sum_{j=0}^\infty \left(\frac1{2^j} - \frac1{3^j}\right)[/math]
- [math]\sum_{i=1}^\infty \frac{5\cdot2^i+6i}{i2^i}[/math]
- [math]\sum_{k=0}^\infty \left(3+\frac1{3^k}\right)[/math]
- [math]\sum_{i=1}^\infty \frac{i^2-1}{i^2+1}[/math]
- [math]\sum_{k=0}^\infty \frac{2^k+3^k}{6^k}[/math].
BBy Bot
Nov 03'24
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[/math]
Consider the infinite series [math]\sum_{i=0}^\infty a_i[/math] defined by
[[math]]
\cond{a_{2i} = \frac1{2^i}, & i = 0, 1, 2, \ldots ,}
[[/math]]
[[math]]
\cond{a_{2i+1} = 0, & i = 0, 1, 2, \ldots .}
[[/math]]
Write out the sum of the first ten terms. Does the series converge? If so, to what value?
BBy Bot
Nov 03'24
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[/math]
Using Theorem \ref{thm 9.2.2}, show that if [math]\sum_{i=m}^\infty a_i[/math] converges and if [math]\sum_{i=m}^\infty b_i[/math] diverges, then [math]\sum_{i=m}^\infty (a_i+b_i)[/math] must diverge.
BBy Bot
Nov 03'24
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[/math]
Is it true that if the series [math]\sum_{i=m}^\infty (a_i+b_i)[/math] converges, then both [math]\sum_{i=m}^\infty a_i[/math] and [math]\sum_{i=m}^\infty b_i[/math] must also converge? Give a reason for your answer.
BBy Bot
Nov 03'24
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[/math]
Prove that the harmonic series [math]\sum_{k=1}^\infty \frac1k[/math] diverges using the following elementary argument. Begin by grouping the terms of the series:
[[math]]
\sum_{k=1}^\infty \frac1k =
1 + \left(\frac12+\frac13\right) +
\left(\frac14+\frac15+\frac16+\frac17\right)
[[/math]]
[[math]]
+ \left(\frac18+\frac19+\frac1{10}+\frac1{11}+\frac1{12}+
\frac1{12}+\frac1{13}+\frac1{14}+\frac1{15}\right) + \cdots
,
[[/math]]
and observe that
[[math]]
\frac12 + \frac13 \gt \frac14 + \frac14 = \frac12
,
[[/math]]
[[math]]
\frac14+\frac15+\frac16+\frac17 \gt
\frac18+\frac18+\frac18+\frac18=\frac12,
\quad \mbox{etc.}
[[/math]]
BBy Bot
Nov 03'24
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[/math]
Consider the infinite series [math]\sum_{k=1}^\infty \left(\frac1k - \frac1{k+1}\right)[/math]. By writing out a few terms of the sequence of partial sums, show that the series converges, and give its value.
BBy Bot
Nov 03'24
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[/math]
An infinite series of the form [math]\sum_{i=m}^\infty (a_i - a_{i+1})[/math] is called a telescoping series (see Problem Exercise). Prove that it converges if and only if the sequence [math]\{ s_n \}[/math] converges. If it does converge, what is its value?
BBy Bot
Nov 03'24
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[/math]
Determine whether or not each of the following infinite series converges, and evaluate it if it does.
- [math]\sum_{k=1}^\infty \frac1{k(k+1)}[/math].
- [math]\sum_{i=1}^\infty \frac{2i+1}{i^2(i^2+2i+1)}[/math].
- [math]\sum_{k=1}^\infty \ln \left(\frac{k+1}k\right) = \ln \left(\frac21\right) + \ln \left(\frac32\right) + \ln \left(\frac43\right) + \cdots [/math].