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Notes on Series - Continued

# Notes on Series - Continued - Notes on Series Continued...

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Notes on Series - Continued Convergent series may be thought of as finite sums as they ‘sum’ to a finite number. Therefore we can have an arithmetic associated with them. Divergent series though do not have arithmetical properties. Convergent series may be multiplied by a constant number, added together, subtracted from each other, multiplied together and divided. In this course we only consider the first three arithmetic operations though. The theorem below provides the relevant statements. Theorem : A) Suppose n a and n b are convergent series and c is a constant. Then the series n ca and ( 29 n n a b ± are all convergent series, and n n ca c a = ( 29 n n n n a b a b ± = ± B) Suppose n a diverges and c is a constant. Then n ca diverges for 0 c , and, if n b is a convergent series, then ( 29 n n a b ± diverges. Examples : Below are some examples of how this theorem may be used to determine convergence/divergence of certain series. 1) Show the series 1 1 1 1 2 2 3 n n n + - = - converges and find the sum of the series.

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