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CalculusBy Educator Today

C48 · Unit C10 Sequences and series

Power-series representations and operations

BC extensionReview editionNot yet human-reviewed
Time
About 45 minutes, untimed
Course framework reference
AP 10.14–10.15 · BC
Builds on
Part of

Progress on this lesson counts in every pathway listed.

Understand the idea

Known series can be substituted, multiplied by powers, differentiated or integrated inside their radius of convergence. Begin with 1/(1 minus x) equals sum sub n ₌ sub 0 ∞x to the power n for |x|<1. Integrating from 0 to x gives minus ln(1 minus x) equals sum sub n ₌ sub 1 ∞x to the power n /n. Differentiation and integration preserve the radius but can change endpoint inclusion. Equality with a Taylor series must be justified by a remainder or a known valid identity.

Worked example

Replace x by minus x squared in the geometric series to obtain 1/(1+x squared) equals sum sub n ₌ sub 0 ∞( minus 1) to the power n x^(2n), |x|<1. Integrating from 0 to x gives arctan x equals sum sub n ₌ sub 0 ∞( minus 1) to the power n x^(2n+1)/(2n+1). The latter also converges at plus or minus 1 by the alternating test.

Explore the mathematics

Power-series representations and operations

Connect power-series operations with a function and its convergence interval.

Setting 3 of 5

Scroll sideways to see the full diagram.

Can power series be differentiated and integrated term by term?xy-2-112eˣ and Taylor degree 3 about 0

Predict before continuing

Can power series be differentiated and integrated term by term?

Read the complete explanation
  1. Can power series be differentiated and integrated term by term?
  2. Within the open convergence interval, the power series represents a function.
  3. Differentiating eˣ=Σxᵏ/k! reproduces the same series.
  4. Operations preserve the radius, but endpoint convergence must be checked again. Finite polynomial behaviour alone is not a convergence proof.
Power-series representations and operations · visual worked method
Power-series representations and operations · visual worked methodReplace x by −x² in the geometric series to obtain 1/(1+x²)=Σₙ₌₀∞(−1)ⁿx^(2n), |x|<1. Integrating from 0 to x gives arctan x=Σₙ₌₀∞(−1)ⁿx^(2n+1)/(2n+1). The latter also converges at ±1 by the alternating test.Worked example · 1/2Replace x by −x² in the geometricseries to obtain1/(1+x²)=Σₙ₌₀∞(−1)ⁿx^(2n), |x|<1.Integrating from 0 to x givesarctanWorked example · 2/2x=Σₙ₌₀∞(−1)ⁿx^(2n+1)/(2n+1). Thelatter also converges at ±1 by thealternating test.

Replace x by −x² in the geometric series to obtain 1/(1+x²)=Σₙ₌₀∞(−1)ⁿx^(2n), |x|<1. Integrating from 0 to x gives arctan x=Σₙ₌₀∞(−1)ⁿx^(2n+1)/(2n+1). The latter also converges at ±1 by the alternating test.

Discuss: explain what each label represents, then draw a related example with different values.

A common misconception

Do not carry endpoint inclusion through differentiation or integration without retesting. The open-radius operation theorem does not settle endpoints.

Investigate and explain

Derive a new function series from the geometric series, then independently check its first three derivatives at zero.
Identify one step in the worked example, name the mathematical fact that permits it, and state its domain conditions.