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STEP Support Programme

Sequences & Series

Hi again everyone!

I've been looking at the STEP specification, I find that it's pretty vague in describing what you actually need to know. I don't know if anyone shares my views in this but this is how I feel. Also, from looking at STEP questions specifically on Sequences & Series, it seems you need to know a lot more than what's on the syllabus.

I would like to go through a list of things and ask for peoples opinion on whether I need to need to know it or not.

Right, the specification says:

Sequences and Series: including use of, for example, $ a_{n+1} = f(a_n) $or$ a_{n+1}= f(a_n, a_{n-1})$; including understanding of the terms convergent, divergent and periodic in simple cases; including use of $ \sum_{k=1}^n k $ to obtain related sums.

Questions: Does understanding of the terms convergent and divergent mean being able to prove a Sequence or a Series is convergent or divergent? How 'formal' does the proof have to be? Would showing that $\lim a_n = L $ be sufficient for showing a sequence converges or would we have to use a $\epsilon-N$ proof technique?

Do we have to know any tests for convergence of a series? I.e. Integral test, comparison test, Alternating Series Test, Ratio test etc...

This was not done on purpose, I thought I was editing my post. Could someone delete this please?

I have left this one as one of our mentors had replied here (it was the top one!) Others have been deleted.

Simple answer... No. Understanding of this sort of thing is not needed in STEP and wouldn't be required to answer any questions.ON the other hand looking at the spec and going beyond it a bit is a good thing to do as learning more maths will broaden your understanding of simpler concepts. in conclusion, if you have the time yes sure learn these kind of things but remember you don't need to know them and questions are designed for you to do without them

Telescoping series (where lots of terms cancel) is not explicitly mentioned, and is often on Further Maths specifications, but can be required in STEP I and II. It's not particularly tricky, but might be off-putting if you have not come across it before.

Best thing to do is to write out the first few terms, until you can see what cancels, and the last term or two.

For example, consider $\sum_{i=1}^n \left(\frac 1 i - \frac 1 {i+1} \right)$. If you write out a few terms you will see that most cancel apart from one at the start and one at the end. You can use this to show that $\sum_{i=1}^n \frac 1 {i(i+1)} =\sum_{i=1}^n \left(\frac 1 i - \frac 1 {i+1} \right)= 1 - \frac 1 {1+n}$.

It is a good idea to test your answers with a few values of $n$, e.g. $n=1,2,3$.

Yes, I know a bit about those.

What would most questions require of me though? Would showing that something like $\sum \frac{1}{i(i+1)} $ converges?

At the level of STEP, I think you'd only be required to spot if something converges if you have (or can derive) an explicit formula for the partial sums (sum up to the $n$th term) e.g. the telescoping sum above, $$\sum_{r=1}^n \frac{1}{r(r+1)}=1-\frac{1}{1+n} \to 1 \text{ as } n \to \infty ,$$ or a geometric series when the common ratio $r$ has $|r|<1$, $$\sum_{r=1}^n a r^{n-1} = \frac{a(1-r^n)}{1-r} \to \frac{a}{1-r} \text{ as } n \to \infty.$$

By something, I mean a series in this case.