4.3 · Arithmetic and geometric sequences and series

A sequence is a function on integer inputs

An arithmetic sequence has constant difference $\displaystyle d$ and formula $\displaystyle a_n=a_1+(n-1)d$. A geometric sequence has constant ratio $\displaystyle r$ and formula $\displaystyle a_n=a_1r^{n-1}$.

The indexing convention matters. If the starting value is labeled $\displaystyle a_0$, the exponent or offset changes. Compare the formula at the stated first index before using it.

A series adds the sequence terms

The finite arithmetic sum is $\displaystyle n(a_1+a_n)/2$. For a geometric sequence with $\displaystyle r\ne1$, the sum is $\displaystyle a_1(1-r^n)/(1-r)$. If $\displaystyle r=1$, simply add $\displaystyle n$ identical terms.

Summation notation specifies both a rule and an index interval. Count the included terms before applying a closed form; an upper index does not always equal the number of terms.

An infinite geometric series needs a shrinking ratio

If $\displaystyle |r|<1$, the geometric partial sums approach $\displaystyle a_1/(1-r)$. If $\displaystyle |r|\geq1$ and the first term is nonzero, the infinite geometric series does not converge.

A sequence approaching zero is necessary for a series to converge but is not sufficient in general. The finite geometric formula remains valid even when an infinite sum is not.

$$S_{\infty}=\frac{a_1}{1-r},\qquad |r|<1$$