Revision:Sequences and Series (A-level)

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These notes are based on the requirements for C1 or C2 A Level mathematics modules.

Sequences and Series

Just a short revision summary for this one - there are more than enough questions in the Heinemann books!

Proof of sum of arithmetic series

(where L is the last term of the series)

Add these two:

Since L is the last term, we know it equals , where n is the number of terms of the series in the sum.

Proof of sum of geometric series

(we have no need for L this time)

Take the first from the second:

Sum of convergent geometric series to infinity. This only happens when -1 < r < 1, because if r is any larger than one (or minus one), rⁿ will tend to infinity rather than zero as n tends to infinity (as it does when you continue the series to infinity!), which will mean there is no sum to infinity.

So we have:

As n tends to infinity, rⁿ tends to zero, so (1 - rⁿ) tends to one, so:

tends to

And this is the sum to infinity of a convergent geometric series.

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Originally written by mik1w on TSR forums.