14-3: Series and Partial Sums
This chapter is all about formulas. The sequence formulas are from 14-2 (they appear in Katie's post), but I will put them here so you can see them in relation to the series formulas.
First of all, a series is the sum of the values of a sequence.
A partial sum is the sum of n terms in the series.
Things to keep in mind:
- n is the term number
- tn is the term value
- Sn is the nth partial sum
- "lim" stands for limit. This means that in a geometric series, if the constant ratio is less than 1, the partial sum of that series will approach a limit. In the geometric partial sum equation, as n gets larger and larger, r^n gets smaller and smaller, closer and closer to 0. Therefore, r^n becomes more insignificant and is "eliminated" from the lim equation. This situation is known as a convergent series because the series converges to a particular value as a limit.
- If in a geometric series |r| > 1, it is called a divergent series because the terms do not go to zero and thus the series diverges.
Q: Find the 10th partial sum of the series 1, 1/2, 1/4, 1/8...
To what limit does the series converge?
r = 1/2 (a common difference), so this is a geometric series
Use the formula for the partial sum of a geometric series.
Then use the limit formula.
Binomial Series Formulas
A binomial series (binomial expansion) is of the form (a+b)^n.
2 Ways to get the coefficients of the expanded series:
- Pascal's Triangle
- Binomial Theorem
Things to keep in mind (in a binomial series) - these can come in helpful when checking your work:
- There are (n+1) terms
- Each term has degree n.
- Powers of a start at a^n and decrease by 1. Power of b start at b^0 and increase by 1.
- Sum of exponents in each term is n.
- Coefficients symmetrical w/ respect to series' ends.
Q: Given (2a-3b)^7, find the term with b^4.
- Write down (-3b)^4, because you know this is your b^4 term.
- Multiply that by (2a)^3, because 2a is your "a" term and the exponents of the entire term must add up to 7. --> 7 - 4 = 3.
- Multiply everything by 12C7, the coefficient of the term.
- 35 * 8 * 81(a^3)(b^4)
Q: In the binomial series (a-b)^17, find the 8th term.
A & Solution: This requires one more step in your thought process. You must figure out what b term the "8th term" will be. You know the first b term is b^0, the second is b^1, the third is b^2, etc. So go forward and the 8th b term is b^7.
- Write down (-b)^7.
- Multiply by a^10 because the exponents of the term must add up to 17.
- Multiply by 17C7, the coefficient of the term.
Formulas, Worked Examples, Pascal's Triangle