series convergence or divergence

Jan 2010
72
2
Does this series diverge or converge:

1 + [1*2]/[1*3] + [1*2*3]/[1*3*5] + [1*2*3*4]/[1*3*5*7} + ...
 
Jul 2009
555
298
Zürich
Does this series diverge or converge:

1 + [1*2]/[1*3] + [1*2*3]/[1*3*5] + [1*2*3*4]/[1*3*5*7} + ...
This is \(\displaystyle \sum_{n=1}^\infty a_n\), where \(\displaystyle a_1=1\), and \(\displaystyle a_n=a_{n-1}\cdot\tfrac{n}{2n-1}\), and, yes, this series converges.
For consider that \(\displaystyle \lim_{n\to\infty}\frac{n}{2n-1}=\frac{1}{2}<1\)
 
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CaptainBlack

MHF Hall of Fame
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Does this series diverge or converge:

1 + [1*2]/[1*3] + [1*2*3]/[1*3*5] + [1*2*3*4]/[1*3*5*7} + ...
\(\displaystyle a_{n+1}=a_n \frac{n+1}{2n+1}\)

so:

\(\displaystyle \frac{a_{n+1}}{a_n}=\frac{1}{2-\frac{1}{n+1}}\)

CB
 
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Prove It

MHF Helper
Aug 2008
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Does this series diverge or converge:

1 + [1*2]/[1*3] + [1*2*3]/[1*3*5] + [1*2*3*4]/[1*3*5*7} + ...
\(\displaystyle 1 + \frac{1 \cdot 2}{1 \cdot 3} + \frac{1 \cdot 2 \cdot 3}{1 \cdot 3 \cdot 5} + \frac{1 \cdot 2 \cdot 3 \cdot 4}{1 \cdot 3 \cdot 5 \cdot 7} + \dots\)

\(\displaystyle = \sum_{k = 1}^{\infty}{\frac{k!}{\frac{(2k)!}{k!2^k}}}\)

\(\displaystyle = \sum_{k = 1}^{\infty}\frac{(k!)^22^k}{(2k)!}\)


Using the ratio test:

\(\displaystyle \lim_{n \to \infty}\left|\frac{t_{n + 1}}{t_n}\right| = \lim_{n \to \infty}\left|\frac{\frac{[(n + 1)!]^22^{n + 1}}{[2(n + 1)]!}}{\frac{(n!)^22^n}{(2n)!}}\right|\)

\(\displaystyle = \lim_{n \to \infty}\left|\frac{(2n)![(n + 1)!]^22^{n + 1}}{[2(n + 1)]!(n!)^22^n}\right|\)

\(\displaystyle = \lim_{n \to \infty}\left|\frac{2(2n)!(n + 1)^2(n!)^22^n}{(2n + 2)(2n + 1)(2n)!(n!)^22^n}\right|\)

\(\displaystyle = \lim_{n \to \infty}\left|\frac{2(n + 1)^2}{2(n + 1)(2n + 1)}\right|\)

\(\displaystyle = \lim_{n \to \infty}\left|\frac{n + 1}{2n + 1}\right|\)

\(\displaystyle = \lim_{n \to \infty}\left(\frac{n + 1}{2n + 1}\right)\)

\(\displaystyle = \lim_{n \to \infty}\frac{1}{2}\) by L'Hospital's Rule

\(\displaystyle = \frac{1}{2}\)

\(\displaystyle < 1\).


So the series is convergent.
 
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CaptainBlack

MHF Hall of Fame
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\(\displaystyle 1 + \frac{1 \cdot 2}{1 \cdot 3} + \frac{1 \cdot 2 \cdot 3}{1 \cdot 3 \cdot 5} + \frac{1 \cdot 2 \cdot 3 \cdot 4}{1 \cdot 3 \cdot 5 \cdot 7} + \dots\)

\(\displaystyle = \sum_{k = 1}^{\infty}{\frac{k!}{\frac{(2k)!}{k!2^k}}}\)

\(\displaystyle = \sum_{k = 1}^{\infty}\frac{(k!)^22^k}{(2k)!}\)


Using the ratio test:

\(\displaystyle \lim_{n \to \infty}\left|\frac{t_{n + 1}}{t_n}\right| = \lim_{n \to \infty}\left|\frac{\frac{[(n + 1)!]^22^{n + 1}}{[2(n + 1)]!}}{\frac{(n!)^22^n}{(2n)!}}\right|\)

\(\displaystyle = \lim_{n \to \infty}\left|\frac{(2n)![(n + 1)!]^22^{n + 1}}{[2(n + 1)]!(n!)^22^n}\right|\)

\(\displaystyle = \lim_{n \to \infty}\left|\frac{2(2n)!(n + 1)^2(n!)^22^n}{(2n + 2)(2n + 1)(2n)!(n!)^22^n}\right|\)

\(\displaystyle = \lim_{n \to \infty}\left|\frac{2(n + 1)^2}{2(n + 1)(2n + 1)}\right|\)

\(\displaystyle = \lim_{n \to \infty}\left|\frac{n + 1}{2n + 1}\right|\)

\(\displaystyle = \lim_{n \to \infty}\left(\frac{n + 1}{2n + 1}\right)\)

\(\displaystyle = \lim_{n \to \infty}\frac{1}{2}\) by L'Hospital's Rule

\(\displaystyle = \frac{1}{2}\)

\(\displaystyle < 1\).


So the series is convergent.
This looks like very hard work compared to the two previous posts, I expect that explains why it took 11 minutes longer to write.

CB
 
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