MATH1061 Chap.4 Series
Series
Taylor polynomials turn local derivative information into a practical approximation. This chapter builds each coefficient from the function's derivatives at a chosen centre, distinguishes the polynomial degree from the number of written terms and shows why the approximation is strongest near that centre.
The remainder is treated as part of the answer rather than an afterthought: it describes what has been omitted and, when a derivative bound is available, supplies a defensible error estimate. A coefficient table keeps the construction transparent by listing derivative order, derivative value at the centre and factorial divisor.
Familiar Maclaurin series can speed up work, but their convergence conditions and sign patterns still need checking. Substituting a new expression changes every power and may change the effective degree. When a numerical approximation is requested, the polynomial value and the error guarantee play different roles: one estimates the target and the other states how far the estimate may be from it.
What this chapter covers
- 01
Build Taylor coefficients from derivatives
- 02
Distinguish centre, degree and evaluation point
- 03
Recognise standard Maclaurin expansions
- 04
Approximate functions near the centre
- 05
Bound error with a remainder term
- 06
Compare lower and higher degree fits
Worked example · free
Approximate an exponential value locally
- 1The degree-three polynomial is 1 + x + x squared/2 + x cubed/6.
- 1Substitute x = 0.2 to obtain 1 + 0.2 + 0.04/2 + 0.008/6.
- 1The approximation is 1.221333 to six decimal places.
- 1The first omitted series term is x to the fourth/24, which is positive at 0.2.
Key terms
- Series centre
- The input where the polynomial matches the function and its selected derivatives.
- Maclaurin series
- A Taylor series centred at zero.
- Polynomial degree
- The highest power of the variable appearing with a nonzero coefficient.
- Remainder term
- The difference between the exact function value and its Taylor polynomial approximation.
- Coefficient
- A constant multiplying a power in a polynomial or series.
- Convergence interval
- The inputs for which an infinite series approaches a finite represented value.
- Geometric ratio
- The fixed multiplier relating each term of a geometric series to the previous term.
- Error tolerance
- The largest approximation difference allowed by a stated numerical requirement.
Series FAQ
Why is a Taylor approximation local?
Its coefficients force the polynomial and function to share values of selected derivatives at one centre. Those matching conditions control nearby behaviour most strongly. Farther away, omitted higher-order terms can grow and the error may increase. A higher degree can help, but the centre and interval still matter when judging reliability.
How do degree and number of terms differ?
Degree names the highest power retained, while the number of visible terms depends on which coefficients vanish. A degree-four polynomial may have fewer than five nonzero terms. State the highest included power explicitly and avoid counting only printed terms, especially for sine or cosine expansions where alternating coefficients are zero.
What should an error statement contain?
Name the approximation, the interval or evaluation point and the remainder information used. A first omitted term can indicate likely scale in a familiar alternating or positive series, but a rigorous Lagrange bound needs a maximum for the relevant higher derivative on the interval between the centre and the evaluation point.
How do I choose the expansion centre?
Choose a centre where derivatives are easy to evaluate and the target input is nearby. If the problem specifies a centre, preserve it in every power as x minus that centre. When you have a choice, moving closer can reduce powers in the remainder bound, but only if the required derivative information remains available and valid on the interval.
Exam move
Make a derivative table before memorising any expansion. Record the function, successive derivatives and their values at the centre; then build the coefficients from that table. Practise changing the centre so the general formula remains active. Keep separate columns for derivative order, factorial and coefficient to expose sign or scaling errors.
Reconstruct standard sine, cosine and exponential patterns from two or three derivatives before using memory. On a substitution into a known series, rewrite every power explicitly and identify the resulting degree. For every approximation, write a final line containing degree, centre, evaluation point and error information. Compute a safe derivative maximum on the whole connecting interval rather than at one convenient point.
Compare two degrees on the same small graph to see how matching more derivatives improves the local fit. If a tolerance is missed, increase the degree or choose a justified closer centre; extra decimal rounding cannot repair approximation error. Finish mixed practice by deciding whether each task asks for a polynomial, an infinite series, a numerical estimate or a rigorous bound.
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