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What is the requirement of a series for the method of differences to be applicable??
The general term, $u_r$ of a series can be expressed in the form $f(r) - f(r+1)$.
If the general term of a series $u_r$ can be expressed as $f(r) - f(r + 1)$, how could you write the series??
$$ \sum^n_{r = 1} (f(r) - f(r + 1)) $$
For a general series $\sum^n_{r = 1} (f(r) - f(r + 1))$, what is $u_1$??
$$ f(1) - f(2) $$
For a general series $\sum^n_{r = 1} (f(r) - f(r + 1))$, what is $u_2$??
$$ f(2) - f(3) $$
For a general series $\sum^n_{r = 1} (f(r) - f(r + 1))$, what is $u_r$??
$$ f(r) - f(r + 1) $$
For a general series $\sum^n_{r = 1} (f(r) - f(r + 1))$, what are the first and last few terms of the series??
$$ \begin{align*} +\ f(1) &- f(2) \\ +\ f(2) &- f(3) \\ +\ f(3) &- f(4) \\ \dots \\ +\ f(n) &- f(n + 1) \ \end{align*} $$
$$\begin{align*} +\ f(1) &- f(2) \ +\ f(2) &- f(3) \ +\ f(3) &- f(4) \ \dots \ +\ f(n) &- f(n + 1) \end{align*}$$ What does this cancel down to??
$$ f(1) - f(n + 1) $$
2021-01-28
If you’re doing a method of differences question with three partial fractions, what is probably true that means they cancel out??
The fractions along the diagonals add up to something that is cancelled out.
$$\frac{1}{3} - \frac{1}{2(n + 1)}$$ What is the value of this expression as $n \to \infty$??
$$ \frac{1}{3} $$
If asked to find the limit of a series after a Method of Differences question, should you combine the fractions or leave them seperated??
Leave them seperated.
$$\frac{1}{2(n + 1)}$$ How would you write what this is equal to in an exam??
As $n \to \infty$
$$ \frac{1}{2(n+1)} \to 0 $$
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date: 2021-01-26 13:40
tags:
- '@?further-maths'
- '@?series'
- '@?public'
title: Further Maths - The Method of Differences
