# Taylor series, convergence and error

> The coefficient of (x−a)^n is f⁽ⁿ⁾(a)/n!, not just the derivative.

Course: [University calculus and multivariable methods](https://studybrix.com/en/courses/university-calculus/index.md)
Level: university
Prerequisites: Limits, derivatives and infinite series.

## Notes

### Coefficients

The coefficient of (x−a)^n is f⁽ⁿ⁾(a)/n!, not just the derivative.

### Radius

The radius of convergence controls an open interval or disk; endpoints require separate tests.

### Remainder

A bound on the next derivative over the relevant interval yields a Taylor remainder bound.

### Alternation

An alternating-series error bound requires terms decreasing in magnitude to zero.

## Practice questions

### 1. Find the coefficient of x³ in the Maclaurin series of exp(2x).

Answer: 4/3

Explanation: The coefficient is 2³/3!=8/6=4/3.

### 2. Find the radius of convergence of the power series sum from n=0 to infinity of (x/4)^n.

Answer: 4

Explanation: This geometric series converges for abs(x/4)<1.

### 3. The power series sum x^n/n, n≥1, has radius 1. What happens at x=1?

A. It diverges
B. The ratio test proves endpoint convergence
C. It equals zero
D. It must converge because the radius is 1

Answer: It diverges

Explanation: At x=1 this is the harmonic series. The radius alone does not settle endpoints.

### 4. Does the series sum from n=1 to infinity of (−1)^n/n converge absolutely, conditionally, or not at all?

A. It converges absolutely
B. It has no terms
C. It diverges because all negative terms diverge
D. It converges conditionally

Answer: It converges conditionally

Explanation: The alternating harmonic series converges but its absolute-value series diverges.

### 5. Approximate exp(0.1) by 1+x+x²/2. Give the polynomial value.

Answer: 1.105

Explanation: Substitution gives 1+0.1+0.005=1.105; this is an approximation to exp(0.1).

### 6. Why does differentiating an infinite series term by term require a theorem rather than just linearity for finite sums?

Sample answer / marking guidance: An infinite series involves a limiting operation. Interchanging that limit with differentiation needs hypotheses, such as those valid inside a power series radius.

Explanation: Award credit for distinguishing finite sums from limits and mentioning applicable convergence or power-series hypotheses.

### 7. For the alternating series 1−1/2+1/3−1/4+..., what upper bound on absolute error follows after the fourth term?

Answer: 0.2

Explanation: The positive term magnitudes decrease to zero, so the next magnitude 1/5 bounds the remainder.

### 8. Use a geometric-series identity to derive 1/(1−x)=sum x^n for abs(x)<1.

Sample answer / marking guidance: The partial sum is (1−x^(N+1))/(1−x). Since x^(N+1) tends to zero, the limit is 1/(1−x).

Explanation: Credit the finite-sum identity, the domain abs(x)<1, and the vanishing remainder.

## Further reading

- https://ocw.mit.edu/courses/18-100a-real-analysis-fall-2020/
- https://ocw.mit.edu/courses/18-06-linear-algebra-spring-2010/

Original AI-assisted educational draft; not independently teacher-reviewed. References support further study, not copied exercises, affiliation or curriculum accreditation.