Sequences, Series and Intro to Limits

Notes

Sequences and series

• Arithmetic: $a_n = a_1 + (n - 1)d$ and $\displaystyle S_n = \frac{n(a_1 + a_n)}{2}$.
• Geometric: $a_n = a_1 r^{n - 1}$ and $\displaystyle S_n = \frac{a_1(1 - r^n)}{1 - r}$. If $|r| < 1$ the infinite sum is $\displaystyle \frac{a_1}{1 - r}$.
• Sigma notation $\displaystyle \sum_{k=1}^{n} a_k$ adds terms. Useful sums: $\displaystyle \sum k = \frac{n(n+1)}{2}$.

Binomial theorem

• $\displaystyle (a + b)^n = \sum_{k=0}^{n}\binom{n}{k}a^{n-k}b^k$, with $\displaystyle \binom{n}{k} = \frac{n!}{k!(n-k)!}$.

Limits

• A limit describes what $f(x)$ approaches as $x$ approaches a value. Substitute if possible; if you get $\displaystyle \frac{0}{0}$, factor or rationalize first.
• A two-sided limit exists only if the left and right limits match. At infinity, compare degrees of the numerator and denominator.