9231 · 1.3
Summation of series flashcards
Revision flashcards for Cambridge 9231 Summation of series (syllabus 1.3). Flip, recall, then mark a real past-paper question.
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What is the formula for $\sum_{r=1}^{n} r$?
$\sum_{r=1}^{n} r = \frac{1}{2}n(n+1)$
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What is the formula for $\sum_{r=1}^{n} r^2$?
$\sum_{r=1}^{n} r^2 = \frac{1}{6}n(n+1)(2n+1)$
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What is the formula for $\sum_{r=1}^{n} r^3$?
$\sum_{r=1}^{n} r^3 = \frac{1}{4}n^2(n+1)^2 = \left( \sum_{r=1}^{n} r \right)^2$
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What is the core principle of the 'method of differences'?
To express the general term $u_r$ of a series as a difference of two consecutive terms of another sequence, i.e., $u_r = f(r) - f(r+1)$. This leads to a 'telescoping sum' where intermediate terms cancel.
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How do you find $\sum_{r=5}^{n} u_r$ if you know the formula for $\sum_{r=1}^{n} u_r$?
You calculate $\sum_{r=1}^{n} u_r - \sum_{r=1}^{4} u_r$. Be careful with the upper limit of the subtracted sum; it's one less than the starting limit of the main sum.
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When using the method of differences, what form should you aim for?
$f(r) - f(r+1)$. Sometimes it might be $f(r) - f(r-1)$ or even $f(r) - f(r+2)$. The key is to find a form that creates cancellation.
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How do you find the sum to infinity, $S_{\infty}$?
First, find the sum to $n$ terms, $S_n$. Then, find the limit of $S_n$ as $n \to \infty$. This is only possible if the series converges.
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A common trap: You are asked to find $\sum_{r=1}^{n} (r+1)(r-2)$. What's the first step?
Expand the brackets first to get a polynomial in $r$: $r^2 - r - 2$. Then you can apply the standard summation formulae to $\sum r^2$, $\sum r$, and $\sum 2$.
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For a series summed by the method of differences, how can you tell if it converges?
After finding $S_n$, examine its terms as $n \to \infty$. If any term involving $n$ tends to a finite limit (often zero), the series converges. For example, if $S_n = 1 - \frac{1}{n+1}$, as $n \to \infty$, $\frac{1}{n+1} \to 0$, so $S_{\infty} = 1$.
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What is the sum of a constant, $\sum_{r=1}^{n} c$?
$nc$. You are adding the constant $c$ to itself $n$ times.
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In a 'Show that' question for method of differences, you're given $\frac{1}{r(r+1)} \equiv \frac{1}{r} - \frac{1}{r+1}$. What should you do?
Even if it seems obvious, you must provide the working to prove the identity, usually by combining the RHS over a common denominator. Don't just assume it's true and proceed.