summation by parts

The following corollaries apply Abel’s lemma to allow estimation of certain bounded sums:

Corollary 1

(Summation by parts)
Let $\{a_{i}\},\{b_{i}\}$ be sequences of complex numbers. Suppose the partial sums of the $a_{i}$ are bounded in magnitude by $h$, that $\sum_{0}^{\infty}|b_{i}-b_{i+1}|$ converges, and that $\lim_{i\to\infty}b_{i}=0$. Then $\sum_{0}^{\infty}a_{i}b_{i}$ converges, and

 $\left|\sum_{0}^{\infty}a_{i}b_{i}\right|\leq h\sum_{0}^{\infty}|b_{i}-b_{i+1}|$

Proof. By Abel’s lemma,

 $\sum_{i=0}^{N}a_{i}b_{i}=\sum_{i=0}^{N-1}A_{i}(b_{i}-b_{i+1})+A_{N}b_{N}$

so that

 $\displaystyle\left\lvert\sum_{i=0}^{N}a_{i}b_{i}\right\rvert$ $\displaystyle=\left\lvert\sum_{i=0}^{N-1}A_{i}(b_{i}-b_{i+1})+A_{N}b_{N}\right% \rvert\leq\sum_{i=0}^{N-1}\left\lvert A_{i}(b_{i}-b_{i+1})\right\rvert+\left% \lvert A_{N}b_{N}\right\rvert$ $\displaystyle\leq h\sum_{i=0}^{N-1}\left\lvert b_{i}-b_{i+1}\right\rvert+h% \left\lvert b_{N}\right\rvert$

The condition that the $b_{i}\to 0$ is easily seen to imply that the sequence $\left\lvert\sum_{i=0}^{N}a_{i}b_{i}\right\rvert$ is Cauchy hence convergent, so that

 $\left\lvert\sum_{i=0}^{\infty}a_{i}b_{i}\right\rvert\leq h\sum_{i=0}^{\infty}% \left\lvert b_{i}-b_{i+1}\right\rvert$

since $b_{N}\to 0$.

Corollary 2

(Summation by parts for real sequences)
Let $\{a_{i}\}$ be a sequence of complex numbers. Suppose the partial sums are bounded in magnitude by $h$. Let $\{b_{i}\}$ be a sequence of decreasing positive real numbers such that $\lim_{i\to\infty}b_{i}=0$. Then $\sum_{1}^{\infty}a_{i}b_{i}$ converges, and $|\sum_{1}^{\infty}a_{i}b_{i}|\leq hb_{1}$.

Proof. This follows immediately from the above, since $|b_{i}-b_{i+1}|=b_{i}-b_{i+1}$.

Title summation by parts SummationByParts 2013-03-22 16:28:10 2013-03-22 16:28:10 rm50 (10146) rm50 (10146) 8 rm50 (10146) Theorem msc 40A05 msc 40D05 partial summation