The Partial Summation Formula for Series of Real Numbers

# The Partial Summation Formula for Series of Real Numbers

We will shortly look at two more very useful tests for determining the convergence of series known as Dirichlet's test and Abel's test. Before we look at these two tests though, we will need first look at the following theorem and corollary.

 Theorem 1: If $(a_n)_{n=1}^{\infty}$ and $(b_n)_{n=1}^{\infty}$ are two sequences of real numbers and $A_n = a_1 + a_2 + ... + a_n$ then $\displaystyle{\sum_{k=1}^{n} a_kb_k = A_nb_{n+1} - \sum_{k=1}^{n} A_k(b_{k+1} - b_{k})}$.
• Proof: Let $(a_n)_{n=1}^{\infty}$ and $(b_n)_{n=1}^{\infty}$ be two sequences of real numbers. Note that $a_k = A_k - A_{k-1}$ where we define $A_0 = 0$. Then:
(1)
\begin{align} \quad \sum_{k=1}^{n} a_kb_k &= \sum_{k=1}^{n} [A_k - A_{k-1}]b_k \\ \quad &= \sum_{k=1}^{n} A_kb_k - \sum_{k=1}^{n} A_{k-1}b_k \\ \quad &= \sum_{k=1}^{n} A_kb_k - \sum_{k=1}^{n} A_kb_{k+1} + A_0b_1 + A_nb_{n+1} \\ \quad &= \sum_{k=1}^{n} A_k(b_k - b_{k+1}) + A_nb_{n+1} \\ \quad &= A_nb_{n+1} - \sum_{k=1}^{n} A_k(b_{k+1} - b_k) \quad \blacksquare \end{align}
 Corollary 1: If $(a_n)_{n=1}^{\infty}$ and $(b_n)_{n=1}^{\infty}$ are two sequences of real numbers and $A_n = a_1 + a_2 + ... + a_n$ then if $\lim_{n \to \infty} A_nb_{n+1}$ converges and $\displaystyle{\sum_{k=1}^{\infty} A_k(b_{k} - b_{k+1})}$ converges then $\displaystyle{\sum_{k=1}^{\infty} a_kb_k}$ converges.
• From Theorem 1 we note that:
(2)
\begin{align} \quad \sum_{k=1}^{\infty} a_k = \lim_{n \to \infty} \sum_{k=1}^{n} = \lim_{n \to \infty} A_nb_{n+1} - \sum_{k=1}^{\infty} A_k(b_{k+1} - b_{k}) \end{align}
• Since the terms on the righthand side of the equation above converge, we have that the lefthand side of the equation also converges. $\blacksquare$