Summation by parts and ditichlets test

Reference no: EM13109932

Suppose {a_n} (n=1 to N) and {b_n} (n=1 to N) are two finite sequences of complex numbers. Let B_k = sum b_n (sum n=1 to k) denote the partial sum of the series sum (b_n) with the convention B_0=0.

a. prove the summation by parts by formula

sum (a_n b_n) (sum goes from n=M to N) = a_N B_N -a_M B_{M-1}- sum( [a_{n+1}-a_n}]B_n (where the sum goes from n=M to N-1)

b. Deduce from this Dirichlet's test for convergence of a series: if the partial sums of a series sum(b_n) are bounded, and {a_n} is a sequence of real numbers that decreases monotonically to 0, then sum (a_n b_n) converges

Reference no: EM13109932

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