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Is the series #\sum_(n=0)^\infty1/((2n+1)!)# absolutely convergent, conditionally convergent or divergent?

use the appropriate test...
I know Root wouldn't work with a factorial, but I got stuck on Ratio, too.

I am at #\stackrel(L)(\infty)|1/(2n+2)|#, what should I put next?

Answer 1

Christopher Agresta

The series (\sum_{n=0}^\infty \frac{1}{{(2n+1)!}}) is absolutely convergent.

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Answer 2

Isaiah Allen

#"Compare it with "sum_{n=0}^oo 1/(n!) = exp(1) = e = 2.7182818...#

#"Each term is equal to or smaller than the"#

#sum_{n=0}^oo 1/(n!) = exp(1) = e = 2.7182818...#

#"All terms are positive so the sum S of the series is between"#

#0 < S < e = 2.7182818....#

#"So the series is absolutely convergent."#

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

Is the series #\sum_(n=0)^\infty1/((2n+1)!)# absolutely convergent, conditionally convergent or divergent?

use the appropriate test... I know Root wouldn't work with a factorial, but I got stuck on Ratio, too. I am at #\stackrel(L)(\infty)|1/(2n+2)|#, what should I put next?

use the appropriate test...
I know Root wouldn't work with a factorial, but I got stuck on Ratio, too.

I am at #\stackrel(L)(\infty)|1/(2n+2)|#, what should I put next?