- #1
Mattofix
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Homework Statement
The sum from n=1 to infinity of n!*2^n*n^(-n)
Homework Equations
The Attempt at a Solution
not a clue.
masnevets said:when you simplify that ratio, you get 2(n/(n+1))^n, whose limit is 2/e.
torresmido said:Alright,
the series you have is n!*2^n*n^-n
this can be simplifyed to become:
An= n!*(2/n)^n
so An+1= (n+1)!*(2/n)^(n+1) "you just replace n by n+1
The ratio test says lim as n goes to infinity of An+1/An
which is the same as:
lim as n aprroaches infiniti of [(n+1)!*(2/n)^(n+1)] / [n!*(2/n)^n]
(n+1)!/n! equals n+1 and (2/n)^(n+1)/(2/n)^(n) equals 2/n
so the limit becomes:
lim as n approaches infiniti of (n+1)*(2/n)
which is the same as:
lim as n approaches infiniti of (2n+2)/n
you can factor out the
The limit becomes:
n*(2+2*n^-1)/n
cancel out the n in the denominator and the one above it:
now you have the limit as n aprroaches infiniti of 2+(2/n) or 2+2*n^-1
which equals 2
Mattofix said:yeah - i know that torresmido is wrong.
i have handed it in already though.
but i got to 2(n/(n+1))^n and then didnt know how to get 2/e as the limit.
thanks for your help guys
torresmido said:Sorry guys ... I see my mistake now.
the limimit is 2/e so the series diverges by the ratio test
The answer to this question depends on the specific series in question. In general, a series converges if the terms of the series approach a finite limit as the number of terms approaches infinity. There are various tests that can be used to determine the convergence of a series, such as the comparison test, the ratio test, and the integral test.
Absolute convergence means that the series converges regardless of the order in which the terms are added. Conditional convergence means that the series only converges if the terms are added in a specific order. For example, the alternating harmonic series is conditionally convergent, while the regular harmonic series is absolutely convergent.
No, a series can only converge to a single value. If a series converges, it means that the terms of the series approach a finite limit, which is the value to which the series converges. However, different series can converge to the same value.
An infinite series is a sum of an infinite number of terms. It can be written in the form of Σan, where n represents the number of terms and an represents the nth term of the series. An infinite series can either converge or diverge, depending on the behavior of its terms.
A series diverges if the terms of the series do not approach a finite limit as the number of terms approaches infinity. There are various tests that can be used to determine the divergence of a series, such as the divergence test, the integral test, and the comparison test.