## Geometric Series – AS Maths Revision – Core Mathematics (C2)

Geometric Series(es?)

An illustration of geometric series. Each of the purple squares is 1/4 of the area of the previous square, with the total area being : \frac{1}{4} \,+\, \frac{1}{16} \,+\, \frac{1}{64} \,+\, \cdots \;=\; \frac{1}{3}. SVG redraw of original image. (Photo credit: Wikipedia)

• Hi, I’m ‘a’, the first term in a geometric series
• You may remember me from such series as the arithmetic series from Core 1
• The difference here is that instead of adding something to get from one term to the next, you multiply by something
• That something is the common ratio, ‘r’
• Thus, the series goes a, ar, ar^2, ar^3…
• So the nth term, if you will, is ar^(n-1)
• Sometimes you’ll be given a and r and asked to work out terms
• Other times you’ll be given some terms and asked to work out a and r
• Other times you’ll be given a or r and asked to work out the other, given a term or two
• Simultaneous equations are sometimes involved (solving by substitution, or even elimination)
• Hopefully the Core 2 examiners won’t get any more inventive than that, though

To Infinity (no further)

• If the common ratio is greater than 1, no sum to infinity for you. Go away.
• Wait, come back. Is it less than -1? If so, go away.
• If you’re still here, your common ratio is between (not including) -1 and 1
• That means it’s a convergent function, so a sum to infinity is possible
• Otherwise, it’s a divergent function, in which case the sum to infinity would literally be infinity, so it’s not worth calculating
• Convergent series get closer and closer and closer and closer (you get the picture) to a number
• You can find out what number your series is aiming at using the formula:
• Sum to infinity = a / (1 – r)
• It’s the first term divided by… 1 minus the common ratio
• And it only works for convergent series! (Ones which get smaller each time)

Proving the Sum to n

• Actually, I can’t be bothered to prove it here (mainly because to do so in ASCII would be rather difficult indeed)
• You do seem to need to be able to prove the formula for the sum of a geometric series to n, though, so it might be worth memorizing
• Anyway, what you get is this… (and it is in the Core 2 formula booklet)
• Sum to n = (a * (1 – r^n)) / (1 – r)… or something like that
• It’s pretty easy to use. You may have to use it in an inequality, too, just as a heads up.