Find the coefficient of x³ in the Maclaurin series of exp(2x).
Answer and explanation
4/3
The coefficient is 2³/3!=8/6=4/3.
Subjects
Series transforms
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Sample exercises
Find the radius of convergence of the power series sum from n=0 to infinity of (x/4)^n.
Answer and explanation
4
This geometric series converges for abs(x/4)<1.
The power series sum x^n/n, n≥1, has radius 1. What happens at x=1?
- It diverges
- The ratio test proves endpoint convergence
- It equals zero
- It must converge because the radius is 1
Answer and explanation
It diverges
At x=1 this is the harmonic series. The radius alone does not settle endpoints.
Does the series sum from n=1 to infinity of (−1)^n/n converge absolutely, conditionally, or not at all?
- It converges absolutely
- It has no terms
- It diverges because all negative terms diverge
- It converges conditionally
Answer and explanation
It converges conditionally
The alternating harmonic series converges but its absolute-value series diverges.
Approximate exp(0.1) by 1+x+x²/2. Give the polynomial value.
Answer and explanation
1.105
Substitution gives 1+0.1+0.005=1.105; this is an approximation to exp(0.1).
Why does differentiating an infinite series term by term require a theorem rather than just linearity for finite sums?
Answer and explanation
An infinite series involves a limiting operation. Interchanging that limit with differentiation needs hypotheses, such as those valid inside a power series radius.
Award credit for distinguishing finite sums from limits and mentioning applicable convergence or power-series hypotheses.
Study notes
Taylor series, convergence and error
The coefficient of (x−a)^n is f⁽ⁿ⁾(a)/n!, not just the derivative.
UniversityTaylor series, convergence and error · revision notes
The coefficient of (x−a)^n is f⁽ⁿ⁾(a)/n!, not just the derivative.
UniversityA Taylor coefficient of an exponential
The requested coefficient is not the nth derivative; divide by n factorial.
UniversityA Taylor coefficient of an exponential · formula reference
The requested coefficient is not the nth derivative; divide by n factorial.
UniversityLaplace transform of a monomial
This transform integrates over t ≥ 0. Substitute a positive s after transforming.
UniversityLaplace transform of a monomial · formula reference
This transform integrates over t ≥ 0. Substitute a positive s after transforming.
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