How do you find the sum of the convergent series 10-5/2+5/8-...? If the convergent series is not convergent, how do you know?

Answer 1

Yes, it is convergent because #abs(r)<1#

#r=(-5/2)/10=-1/4#
#abs(-1/4)<1#, so it is convergent
#"infinite geometric sum" = a_1/(1-r)=10/(1-(-1/4))=8#

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Answer 2

To find the sum of the convergent series (10 - \frac{5}{2} + \frac{5}{8} - \ldots), we need to determine if it follows a specific pattern or formula. This series appears to be an alternating series, where the terms alternate between positive and negative. One common method to find the sum of such series is by using the formula for the sum of an infinite geometric series, given by:

[S = \frac{a}{1 - r}]

Where (a) is the first term of the series and (r) is the common ratio between consecutive terms. In this case, the first term (a = 10) and the common ratio (r = -\frac{1}{4}) (since each term is multiplied by (-\frac{1}{2}) to get the next term).

Substituting these values into the formula:

[S = \frac{10}{1 - \left(-\frac{1}{4}\right)} = \frac{10}{1 + \frac{1}{4}} = \frac{10}{\frac{5}{4}} = \frac{40}{5} = 8]

Therefore, the sum of the given convergent series is 8.

If a series is not convergent, it means that its terms do not approach a finite value as the number of terms approaches infinity. One common method to determine if a series is convergent is by checking the behavior of its terms as (n) approaches infinity. If the terms do not approach zero, the series diverges. Additionally, there are various convergence tests, such as the ratio test, the root test, and the alternating series test, which can be used to determine the convergence or divergence of a series.

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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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