Series Basics
An infinite series is not just a list of numbers—it is the limit of their running totals. In this section, we cover the notation, basic arithmetic properties, and how to manipulate indices.
The Sequence of Partial Sums
Given a sequence , the Partial Sum is the sum of the first terms.
The infinite series is defined as the limit of these partial sums:
Visualizing Terms vs. Sums
The terms (Grey) approach 0, but the Sum (Blue) approaches 2.
Vital Distinction
Do not confuse the Sequence (a list of numbers) with the Series (a single value, if it converges).
Properties & Arithmetic
Linearity
If and converge, we can split sums and pull out constants.
Multiplication Warning
Series multiplication is NOT term-by-term. You cannot just multiply the sums.
Multiplication requires distributing every term (Cauchy Product), which is much more complex.
Index Shifting
Sometimes we need a series to start at instead of . We can shift the index as long as we compensate in the terms.
The See-Saw Rule
If you decrease the starting index by , you must increase every in the formula by .
Example 1: Geometric Shift
Start at instead of
Index -1
Terms +1 (replace with )
Example 2: Complex Term
Start at instead of
Stripping Terms
We can "strip out" the first few terms of a series to change the starting index without changing the formula. This is useful when checking convergence.
The convergence of a series depends only on the "tail" (the infinite part). Stripping a finite number of terms does not affect convergence.
Worked Examples
Example 1: Calculating Partial Sums
Calculate the first three partial sums for .
Example 2: Applying an Index Shift
Rewrite to start at .
Shift Start Down by 2
Since we decrease start (), we must increase terms ().
Practice Quiz
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