How do you test the series #Sigma n/((n+1)(n^2+1))# from n is #[0,oo)# for convergence?

Answer 1

To test the series ( \sum \frac{n}{(n+1)(n^2+1)} ) for convergence, you can use the ratio test.

  1. Apply the ratio test by computing the limit of the absolute value of the ratio of successive terms as ( n ) approaches infinity:

    [ \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| ]

    where ( a_n = \frac{n}{(n+1)(n^2+1)} ).

  2. Compute ( a_{n+1} ) and ( a_n ) separately:

    [ a_{n+1} = \frac{n+1}{(n+2)((n+1)^2+1)} ] [ a_n = \frac{n}{(n+1)(n^2+1)} ]

  3. Take the absolute value of the ratio of successive terms:

    [ \left| \frac{a_{n+1}}{a_n} \right| = \frac{\frac{n+1}{(n+2)((n+1)^2+1)}}{\frac{n}{(n+1)(n^2+1)}} ]

  4. Simplify the expression and compute the limit as ( n ) approaches infinity.

  5. If the limit is less than 1, the series converges. If the limit is greater than 1 or the limit does not exist, the series diverges.

By following these steps, you can determine the convergence or divergence of the given series.

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Answer 2

The series:

# sum_(n=0)^oo n/((n+1)(n^2+1))#

is convergent.

You can test it by direct comparison, considering that for #n>1#:
#0 < n/((n+1)(n^2+1)) < n/n^3 = 1/n^2#

As the series:

#sum_(n=0)^oo 1/n^2#

is convergent based on the p-series test, then also the series:

# sum_(n=0)^oo n/((n+1)(n^2+1))#

is 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.

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.

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.

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.

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