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If the upper limit of a summation is infinite, does it even have a sum?

label Algebra
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I've been given a series in summation notation. The lower limit is n=1 and the limit above the sigma is infinity. I'm asked to write the first four terms and if the series diverges or converges (and I understand those parts). But then it asks me for the to find the sum of the series. How can it have a sum if it goes on infinitely?

Oct 17th, 2017

For a series ∑n=1 an consider the sequence of its partial sums s1 = a1, s2 = a1 +a2, s3 = a1 + a2 + a3, ... ,

sn = ∑k=1n ak , … . The series converges if and only if the sequence sn converges and the sum of the series is the limit of the sequence sn.

Example. The series ∑n=1 (1/2)n converges and its sum is 1 because sn = ∑k=1n (1/2)k = 1/2 + 1/4 + 1/8 + ... +(1/2)n = 1 – (1/2)n → 1 as n → ∞.


Apr 12th, 2015

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Oct 17th, 2017
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