Title: InkMaths / Sequences and series
On every note of this deck, unless a note says otherwise:
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Front: RECALL
State the validity condition for the general binomial expansion.
Back: With n negative or fractional the expansion holds only for |x| < 1. Take out a to the n from (a + bx) first, and the condition becomes |bx/a| < 1. Quoting |x| < 1 for (2 + 3x) to the −1 is out by a factor of three.

Front: SPOT THE ERROR
Say what is wrong with this claim:
The series 2 + 3 + 4.5 + 6.75 + … has sum to infinity 2/(1 − 1.5) = −4.
Back: Here r = 1.5, so the terms grow and the series diverges. The sum to infinity formula only applies when |r| is below 1. A series of positive terms returning a negative total is the tell, so check r before you use the formula, not after.

Front: STANDARD RESULT · LEARN IT
nth term:
arithmetic u_n = ?, geometric u_n = ?\(\text{arithmetic } u_{n} = ?, \text{geometric } u_{n} = ?\)arithmetic u_n = ?, geometric u_n = ?
Back: a + (n - 1)d, ar^n-1\(a + (n - 1)d, ar^{n-1}\)a + (n - 1)d, ar^(n - 1)

Front: RECALL
State how to pick out one coefficient from a binomial expansion.
Back: Do not expand the lot. The general term of (a + bx) to the n is nCr times a to the (n − r) times (bx) to the r, so set r to the power of x you want and evaluate one term. Raise the whole of bx to the r, since forgetting to raise b is where the mark goes.

Front: DEFINE
Define: periodic sequence.
Back: A sequence that repeats itself after a fixed number of terms, so u of n plus k equals u of n for every n. Its order is the smallest k that does this, since every multiple of that k repeats as well.