Geometric Series Part II

Because timesing is a funny operation, a scaling up or down operation - repeated addition or however you like to look at it, it does strange things to series.

A common ratio of 2, like in the previous post, gives you a series which just keeps getting more and more massive, into infinity. It therefore has no sum - because it just keeps going on. You can find the sum of the first n terms of course.

But what if you had a common ratio of 1? Well then your series stays exactly the same

2,2,2,2,2,2,2,2,2,.....into infinity. It's the same sort of thing as above. Finding a complete sum of the series isnot going to be possible, because there will simply be an infinite load of 2s.

But if your common ratio is a fraction:

1/2, 1/4, 1/8, 1/16

Here the common ratio is 1/2. The terms of this sequence get smaller all the time, but will never reach 0.

When we think of the sequence properly as a series:

1/2 + 1/4 + 1/16 + 1/32 + 1/64

Then we can see that it is an infinite series, and will go on halving ad infinitum.
If we add the first few terms, we get 1/2, 3/4, 13/16, 27/32....

The sum is getting larger each time but by progressively smaller amounts.

The sum is getting closer and closer to 1, but without ever quite reaching it. If it had an infinite number of terms, then its sum would indeed reach 1. Its sum to infinity is 1.

It converges on 1, and is therefore called a convergent series.

This happens with certain types of common ratio, which we will come back to.

Geometric Series

These are a bit like arithmetic series but instead of there being a common difference between terms there is a common power-type difference.

Consider 2,4,8,16,32,64...

This series appears to double each time, which means that there is no common number which gives you the nth term when added to the n-1th term.

But the common difference is "doubling" - or rather, powers of 2.

The series goes 2¹, 2², 2³, 2⁴.....

With geometric series, therefore, we don't talk about a common difference, but a common ratio - the thing you times each term by to get the next one. So this series has common ratio 2.

Sum of an Arithmetic Series

This was supposedly demonstrated by Gauss, aged eight or something, when he was set a problem by a teacher desperate to get on with something more interesting to add together all the numbers from 1-100. He's supposed to have realised that if you paired up the numbers, they all had the same sum ie 101. It was then a matter of spotting the number of pairs and multiplying the two (5050).

The sum of an arithmetic series is done in a similar way. Think of a series as being:

a₁ + (a₁ + d) + (a₁ + 2d) +......

since d is a constant...

You want to find the sum of the first n terms.

S_n = a₁ + (a₁ + d) + (a₁ + 2d) +....

....(a₁ + (n-1)d)

This is probably quite a few terms (otherwise you wouldn't bother trying to find the sum, would you?).

So the wisest option would be to pair up the terms, like Gauss did, and find the sum of each pair.

The formula is derived from writing the above series out from beginning to n, and the opposite way, and adding the terms.

However, just as Gauss's addition always gave him 101, in our case we will always get 2a + (n-1)d

Since we want the sum to n of the series, there will be n of these pairs. So the sum is n(2a + (n-1)d)

BUT -

We have just added TWO series! Because we wrote it out twice to make adding the terms easier.

Therefore, the final formula will be: n(2a + (n-1)d)/2.


Quite fiddly to prove, but easy enough to use.

Series and Sequences 1 - nth terms

There are a few of these.

Let's just start with a definition. A sequence is a load of numbers in a list, with there being a common difference between the numbers.

A series is the same thing, but added together.

Here is an arithmetic sequence: 0,2,4,6,8,10....

Here is an arithmetic series: 0+2+4+6+8+10.....

In an arithmetic series or sequence, there is always a common difference between the numbers. In the ones above, the common difference is 2. It is always a constant (and not anything weird like n² - that's more like geometric series).

So finding the nth term of an arithmetic series or sequence is easy enough.

You need the first term a₁ and the common difference d.

a_n = a₁ + (n-1)d

So say I wanted to find the 42nd term of the above sequence.

a₄₂ = 0 + 41d
= 0 + 41x2
= 82

The 42nd term of the sequence 0,2,4,6,8...is 82


Because the difference is a constant with arithmetic sequences, this formula is simple enough to grasp. To find the nth term you need the first term, and then the number of terms before the nth one timesed by the common difference because there are that many lots of the common difference.

Slowness

Hmm, 50 posts in over a year isn't much is it? Well, I get bored easily.

You should see the other place...

I will try to post more, as I come back to maths after a few months' absence and try to tackle C3.

Radians

These are little blighters used to describe the angles of circles. Like degrees, they can be used to express arc widths, or can be used in equations (especially trig equations). Basically they're just like degrees, but are based on π and the radius of the circle (hence radian). Imagine a line from the centre to any point on the circle. That's a radius. Then, in any direction, draw an arc around the surface of the circle with the same length as the radius. Stop. Draw a radius from this point back to the centre.

The angle subtended is one radian.

There are 360 degrees in a circle and one degree really isn't very large.

A radian is much bigger (around 57 degrees). Every circle has 2π radians, like every circle has 360°. 180° is therefore π radians.

Most calculations involving radians will also involve fractions. Using π enables you to be exact without lots of fiddly decimals. The same is true of fractions. In some ways then, using radians is satisfyingly precise.

It is also easy to convert one to the other.

360° = 2π radians

Therefore

1° = 2π radians/360.

and

say

12° = (2π/360) x 12

= 24π/360

= π/15

and it's usually sufficient to leave it like this for C2. At other levels you might want to do the calculation to n dps or whatever.

Similarly

2π radians = 360°

1 radian = 360/2π

and

so

3 radians = (360/2π) x 3.

Easy.