1.6 Series

For a positive integer n, (a+b)^n=Σᵣ C(n,r)a^{n−r}b^r. The general term identifies the power and coefficient without writing every term.

Track powers of the variable and signs carefully; for a coefficient question, set the target power before simplifying. The expansion is finite only for non-negative integer n.

The x² term in (1+2x)^5 is C(5,2)·(2x)²=40x².

The coefficient C(n,r) is not the whole term; the powers of both factors still contribute.

An arithmetic progression has constant difference d, with uₙ=a+(n−1)d and Sₙ=n/2[2a+(n−1)d]. A geometric progression has constant ratio r, with uₙ=arⁿ⁻¹ and Sₙ=a(rⁿ−1)/(r−1) when r≠1.

Check consecutive differences or ratios before choosing a formula. For an infinite geometric sum, convergence requires |r|<1 and S∞=a/(1−r).

For 3,6,12,…, a=3 and r=2, so it has no finite infinite sum because |r|>1.

A sequence can have a convergent partial-sum pattern without being geometric; never infer r from non-consecutive terms alone.

Use uₙ=a+(n−1)d for an arithmetic progression and uₙ=arⁿ⁻¹ for a geometric progression. Their finite sums follow from adding terms or multiplying by the common ratio.

Identify a,d or r from the information given, keep n as the term number, and check whether a sum is finite or infinite before choosing Sₙ or S∞.

For 5,8,11,…, uₙ=3n+2 and Sₙ=n(3n+7)/2; the first term is included once, not n times.

The nth term and nth partial sum are different quantities: Sₙ is not obtained by substituting n into uₙ.

For first term a and common ratio r, S∞=a/(1−r) exists only if |r|<1. The partial sums approach a finite limit because later terms shrink to zero.

Check convergence before using the formula. A negative r gives alternating partial sums, but still converges when |r|<1.

3−1.5+0.75−… has a=3,r=−0.5 and S∞=3/1.5=2; the alternating signs do not prevent convergence.

A ratio close to 1 may converge slowly, while r=1 or −1 does not produce a finite infinite sum.