Calculus II · Unit 4B · lesson
Power Series as Functions
Treat a power series as an input-dependent numerical series and explore how partial sums behave inside, on, and outside its convergence region.
Section overview
Power-series convergence and endpointsWhat this section is building
Treat a power series as an input-dependent numerical series and explore how partial sums behave inside, on, and outside its convergence region.
Distance from the center organizes the automatic interior and exterior behavior; endpoints remain independent decisions.
Use a ratio or root argument for the radius, convert it to an interval, and test both endpoints explicitly.
Including or excluding both endpoints from the radius calculation alone.
Learning objectives
interpret a power series as a function and evaluate it at specific inputs.
Power Series as Functions
An infinite polynomial is still only meaningful where it converges
A power series resembles a polynomial with infinitely many terms. For each fixed input , however, it becomes an ordinary numerical series. Its value exists only if that numerical series converges. The same symbolic expression may therefore define a function on an interval and fail completely outside it.
The center organizes the powers . Near the center, these powers are small and convergence is favored. Farther away, they grow and can overwhelm the coefficients. Evaluating a power series begins by substituting the input, simplifying the resulting numerical series, and then using the convergence tools from Unit 4A.
One infinite formula can define an entire function
A numerical series asks whether one fixed list of numbers has a finite total. A power series inserts a variable into the terms, so convergence can change when changes. The same expression may converge rapidly at one input, converge delicately at another, and diverge immediately elsewhere.
This is why a power series should be viewed as both an infinite series and a function. First choose an input ; then the power series becomes an ordinary numerical series. The set of inputs where that numerical series converges is the function's domain of representation.
Read this graph as text
A power series changes character when x changes. Three panels show geometric partial sums for x=1/2 approaching 2, for x=-1 oscillating, and for x=2 growing rapidly. Partial sums of the geometric series at an interior, endpoint, and exterior input. Preserve the misconception control described in the lesson and storyboard.
Written labels, distinct line styles, markers, and fill patterns communicate every relationship in a power series changes character when x changes; color is never the only cue.
Why it matters: Partial sums of the geometric series at an interior, endpoint, and exterior input.
Three panels show geometric partial sums for x=1/2 approaching 2, for x=-1 oscillating, and for x=2 growing rapidly.
A power series changes character when x changes. Partial sums of the geometric series at an interior, endpoint, and exterior input.
Power series centered at a
A power series has the form
For each fixed , this becomes a numerical series whose convergence must be determined.
Power-series form
A power series centered at is
At , every positive-power term vanishes, so the value is .
Evaluate at several inputs
For
we have , , and at the terms do not approach zero, so the series diverges.
The same power series behaves three different ways
Consider
At , it is geometric with ratio and sums to . At , the terms alternate between and , so they do not approach zero and the series diverges. At , the terms grow and the series diverges even more plainly. The expression therefore defines only for .
An algebraic formula does not erase the convergence domain
The identity is valid where the geometric series converges, not at every point where the rational function itself exists.
u4b-power_series_as_functions-01Evaluate .
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It is geometric with first term and ratio .
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Evaluate at .
Identify the center of .
Explain why a power series is not automatically defined for every real .
Find the value at the center without summing.
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