Abel's Lemma/Formulation 2

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Lemma

Let $\left \langle {a} \right \rangle$ and $\left \langle {b} \right \rangle$ be sequences in an arbitrary ring $R$.

Let $\displaystyle A_n = \sum_{i \mathop = m}^n {a_i}$ be the partial sum of $\left \langle {a} \right \rangle$ from $m$ to $n$.


Then:

$\displaystyle \sum_{k \mathop = m}^n a_k b_k = \sum_{k \mathop = m}^{n - 1} A_k \left({b_k - b_{k + 1} }\right) + A_n b_n$


Note that although proved for the general ring, this result is usually applied to one of the conventional number fields $\Z, \Q, \R$ and $\C$.


Corollary

$\displaystyle \sum_{k \mathop = 0}^n a_k b_k = \sum_{k \mathop = 0}^{n - 1} A_k \left({b_k - b_{k + 1} }\right) + A_n b_n$


Proof 1

Proof by induction:

For all $n \in \N$ where $n \ge m$, let $P \left({n}\right)$ be the proposition:

$\displaystyle \sum_{k \mathop = m}^n a_k b_k = \sum_{k \mathop = m}^{n - 1} A_k \left({b_k - b_{k + 1} }\right) + A_n b_n$


Basis for the Induction

First consider $P(m)$.

When $n = m$, we have that:

$\displaystyle \sum_{k \mathop = m}^{n - 1} A_k \left({b_k - b_{k + 1} }\right) = 0$

is a vacuous summation, as the upper index is smaller than the lower index.

We also have that:

$\displaystyle A_m = \sum_{i \mathop = m}^m {a_i} = a_m$

Thus we see that $P(m)$ is true, as this just says:

$a_m b_m = 0 + A_m b_m = a_m b_m$

which is clearly true.


This is our basis for the induction.


Induction Hypothesis

Now we need to show that, if $P \left({r}\right)$ is true, where $r \ge m$, then it logically follows that $P \left({r + 1}\right)$ is true.


So this is our induction hypothesis:

$\displaystyle \sum_{i \mathop = m}^r a_k b_k = \sum_{k \mathop = m}^{r - 1} A_k \left({b_k - b_{k + 1} }\right) + A_r b_r$


Then we need to show:

$\displaystyle \sum_{k \mathop = m}^{r + 1} a_k b_k = \sum_{k \mathop = m}^r A_k \left({b_k - b_{k + 1} }\right) + A_{r + 1} b_{r + 1}$

where:

$\displaystyle A_{r + 1} = \sum_{i \mathop = m}^{r + 1} {a_i}$


Induction Step

This is our induction step:

\(\displaystyle \sum_{k \mathop = m}^{r + 1} a_k b_k\) \(=\) \(\displaystyle \sum_{k \mathop = m}^r a_k b_k + a_{r + 1} b_{r + 1}\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \sum_{k \mathop = m}^{r - 1} A_k \left({b_k - b_{k + 1} }\right) + A_r b_r + a_{r + 1} b_{r + 1}\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \sum_{k \mathop = m}^{r - 1} A_k b_k - \sum_{k \mathop = m}^{r - 1} A_k b_{k + 1} + A_r b_r + a_{r + 1} b_{r + 1}\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \sum_{k \mathop = m}^r A_k b_k - \left({\sum_{k \mathop = m}^r A_k b_{k + 1} - A_r b_{r + 1} }\right) + a_{r + 1} b_{r + 1}\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \sum_{k \mathop = m}^r A_k \left({b_k - b_{k + 1} }\right) + \left({A_r + a_{r + 1} }\right) b_{r + 1}\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \sum_{k \mathop = m}^r A_k \left({b_k - b_{k + 1} }\right) + A_{r + 1} b_{r + 1}\) $\quad$ $\quad$

So $P \left({r}\right) \implies P \left({r + 1}\right)$ and the result follows by the Principle of Mathematical Induction.


Therefore:

$\displaystyle \forall n \ge m: \sum_{k \mathop = m}^n a_k b_k = \sum_{k \mathop = m}^{n - 1} A_k \left({b_k - b_{k + 1} }\right) + A_n b_n$

$\blacksquare$


Proof 2

First note that:

$\displaystyle A_{m - 1} = \sum_{i \mathop = m}^{m - 1} a_i = 0$

is a vacuous summation, as the upper index is smaller than the lower index.

Then we have:

\(\displaystyle \sum_{k \mathop = m}^n a_k b_k\) \(=\) \(\displaystyle \sum_{k \mathop = m}^n \left({A_k - A_{k - 1} }\right) b_k\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \sum_{k \mathop = m}^n A_k b_k - \sum_{k \mathop = m}^n A_{k - 1} b_k\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \sum_{k \mathop = m}^n A_k b_k - \sum_{k \mathop = m-1}^{n - 1} A_k b_{k + 1}\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \sum_{k \mathop = m}^{n - 1} A_k \left({b_k - b_{k + 1} }\right) + A_n b_n - A_{m - 1} b_m\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \sum_{k \mathop = m}^{n - 1} A_k \left({b_k - b_{k + 1} }\right) + A_n b_n - 0\) $\quad$ $\quad$

Therefore:

$\displaystyle \forall n \ge m: \sum_{k \mathop = m}^n a_k b_k = \sum_{k \mathop = m}^{n - 1} A_k \left({b_k - b_{k + 1} }\right) + A_n b_n$

$\blacksquare$


Source of Name

This entry was named for Niels Henrik Abel.