How do you test the alternating series #Sigma (-1)^(n+1)n/(10n+5)# from n is #[1,oo)# for convergence?

Answer 1

Diverges by the Divergence Test.

The Alternating Series Test tells us that if we have a series #sum(-1)^nb_n#, where #b_n# is a sequence of positive terms, the series converges if
a) #b_n>=b_(n+1)#, IE, the sequence is ultimately decreasing for all #n.#
b) #lim_(n->oo)b_n=0#
We should note that we don't need to have #(-1)^(n+1)# -- any term that causes alternating signs, such as #cos(npi), (-1)^(n-1), (-1)^(n+1)#, is okay.
Here, we see #b_n=n/(10n+5)#. Taking the limit,
#lim_(n->oo)n/(10n+5)=1/10 ne 0# -- the Alternating Series Test is inconclusive here.

However, take the limit of the overall sequence:

#lim_(n->oo)(-1)^(n+1)n/(10n+5) ne 0# -- we can say this because although the limit does not truly exist, we can convince ourselves that it alternates signs and gets closer to #1/10,# causing divergence by the Divergence Test.
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Answer 2

To test the alternating series ( \sum_{n=1}^{\infty} \frac{(-1)^{n+1}n}{10n+5} ) for convergence, you can use the Alternating Series Test.

  1. Check the terms ( \frac{n}{10n+5} ) to see if they are:

    • Positive
    • Decreasing (strictly decreasing)
    • Tending to zero as ( n ) approaches infinity.
  2. Verify the alternating condition:

    • ( (-1)^{n+1} ) alternates sign.

If both conditions are met, then the series converges.

For the given series:

  1. The terms ( \frac{n}{10n+5} ) are positive, decreasing, and tend to zero as ( n ) approaches infinity because as ( n ) increases, the ( 10n ) term dominates the ( 5 ) term in the denominator, making the fraction tend to ( \frac{n}{10n} = \frac{1}{10} ), which tends to zero.

  2. The series alternates sign with ( (-1)^{n+1} ).

Therefore, the alternating series ( \sum_{n=1}^{\infty} \frac{(-1)^{n+1}n}{10n+5} ) converges by the Alternating Series Test.

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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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