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i have read different articles on Convergence vs Divergence.How the general formula for the partial sums of the series here and here be calculated?

Please let me know how ? Thanks sabbir

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For the second, this series is geometric, and the formula for finding a partial sum of a geometric series is a(1-r^n)/(1-r) where r is the term being raised to a power of i, and n is the number of terms you evaluate to, and a is the coefficient of the sum, which in this case is 1. So, in the second case, r is 1/3, a is 1, and n is unspecified. With some algebra you can turn (1-1/3^n)/(1-1/3) into the solution they give you. One important step is to make sure i=0, which you can do by changing the exponent from i-1 to just i.

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