Series

Definitions #

Infinite sequence: function of which the domain is the set of positive integers and the range is a set of real numbers

$$ \{a_n\}_{n=1}^\infty \text{ or } \{a_n\} $$

Infinite series: the sum of an infinite sequence $$ \sum_{k=1}^\infty a_k \text{ or } \sum a_k $$

Partial sum: $$ S_n = a_1 + a_2 + a_3 + ... + a_n = \sum_{k=1}^n a_k $$

Convergent/divergent series #

• Convergent: If $ \lim_{n \to \infty} a_n = L $ does exsist

• Divergent: If $ \lim_{n \to \infty} a_n = L $ does not exsist

If $S_n$ is convergent then series is convergent and has sum: $$ \lim_{n \to \infty} S_n = \sum_{k=1}^\infty a_k = S $$ Otherwise, if the partial sum is divergent then the serries is divergent.

Special series #

Geometric series #

$$ \sum_{n=1}^\infty a r^{n-1} = a + ar + ar^2 + ... $$

• Convergent: $|r| < 1$ with sum $S = \frac{a}{1-r}$
• Divergent: $|r| \geq 1$

Harmonic series #

$$ \sum_{n=1}^\infty \frac{1}{n} = 1 + \frac{1}{2} + \frac{1}{3} + ... + \frac{1}{n} + ... $$

$\lim_{n \to \infty} \frac{1}{n} = 0$

Tests #

nth term test #

If $\sum_{n=1}^\infty a_n$ is convergent, then $\lim_{n \to \infty} a_n = 0$. (only in this order)

If $\lim_{n \to \infty} a_n \neq 0$ or does not exsist, the series is divergent.

Note that if $\lim_{n \to \infty} a_n = 0$, you can't conclude that the series converges!

Integral test #

If $a_k = f(k)$ for all positive integers $k$ and $n$ is any positive integer (often 1) then $\sum_{k=n}^\infty a_k$ converges if and only if $\int_n^\infty f(x) dx$ exsists.

Ordinary/direct comparison test #

Suppose $0 \leq a_n \leq b_n$:

• If $\sum b_n$ converges, then $\sum a_n$ converges
• If $\sum a_n$ diverges, then $\sum b_n$ diverges

Limit comparison test #

$$ \lim_{n \to \infty}\frac{a_n}{b_n}=c $$

• If $c > 0$ and then either both series converge or diverge.
• If $c = 0$ and $\sum b_n$ converges then $\sum a_n$ converges

Ratio test #

$$ \lim_{n \to \infty} \frac{a_{n+1}}{a_n} = \rho $$

• If $\rho < 1$, the series converges
• If $\rho > 1$, the series diverges
• If $\rho = 1$, the test is inconclusive

Taylor series #

Approximation of graphs by polynomials Why? For making computations / approximations faster. More powers makes the approximation more accurate.