How do you use the limit comparison test to determine if #Sigma n/((n+1)2^(n1))# from #[1,oo)# is convergent or divergent?
Rewriting this:
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To use the limit comparison test to determine if the series ( \sum_{n=1}^{\infty} \frac{n}{(n+1)2^{n1}} ) is convergent or divergent:

Choose a series ( \sum_{n=1}^{\infty} b_n ) that is known to converge and is positive for all ( n ).

Calculate the limit: [ \lim_{n \to \infty} \frac{a_n}{b_n} ] where ( a_n = \frac{n}{(n+1)2^{n1}} ).

If the limit is a positive finite number, then both series either converge or diverge together. If the limit is zero or infinity, then the comparison is inconclusive.

Based on the comparison, if ( \sum_{n=1}^{\infty} b_n ) converges, then ( \sum_{n=1}^{\infty} a_n ) also converges. If ( \sum_{n=1}^{\infty} b_n ) diverges, then ( \sum_{n=1}^{\infty} a_n ) also diverges.
In this case, let's choose ( b_n = \frac{1}{2^n} ) since it's a geometric series that is known to converge.
Then, we calculate the limit: [ \lim_{n \to \infty} \frac{\frac{n}{(n+1)2^{n1}}}{\frac{1}{2^n}} ] [ = \lim_{n \to \infty} \frac{2n}{(n+1)} ]
Now, simplify the limit: [ = \lim_{n \to \infty} \frac{2}{1 + \frac{1}{n}} ] [ = 2 ]
Since the limit is a positive finite number, and ( \sum_{n=1}^{\infty} \frac{1}{2^n} ) converges (it's a geometric series with ( r = \frac{1}{2} < 1 )), by the limit comparison test, ( \sum_{n=1}^{\infty} \frac{n}{(n+1)2^{n1}} ) also converges.
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When evaluating a onesided 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.
When evaluating a onesided 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.
When evaluating a onesided 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.
When evaluating a onesided 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.
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