Find the first five partial sums of the given series and determine whether the s

Question

Find the first five partial sums of the given series and determine whether the series appears to be convergent or divergent. If it is convergent, find its approximate sum. 1/10 - 1/100 - 1/1000 - 1/10000 - 1/10000 - ... S1 = (Type an integer or decimal rounded to four decimal places as needed.) S2 = (Type an integer or decimal rounded to four decimal places as needed.) S3 = (Type an integer or decimal rounded to four decimal places as needed.) S1 = (Type an integer or decimal rounded to four decimal places as needed.)

Explanation / Answer

Ok, I think you can add/subtract here. S1 = sum of first number, S2 = sum of first 2 numbers and so on.

NOw the big question is does it converge or diverge.

Notice it is alternating, which means (-1)^n is needed.

Each term is multiplying by 1/10, so this is a geometric series.

Sum 1/10 (-1/10)^n from n = 0 to infinity.

Now a geometric series converges if in the form r^n and -1<r<1, and in this case r = -1/10 and it Converges!

So the sum is a/(1-r), where a = first term.

In this case a/(1-r) = 1/10 / (1 - -1/10) = 1/10 / (1 + 1/10) = 1/10 / 11/10 = 1/11 (answer).

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