Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2004 / Paper 2 / 8 / b

Table of contents
- i b
  - Solution i
- ii b
  - Solution ii

i

Solution

Multiply the two power series and collect equal powers of

x

. The Cauchy product gives

c_{0} = a_{0} b_{0}, c_{1} = a_{0} b_{1} + a_{1} b_{0}, c_{2} = a_{0} b_{2} + a_{1} b_{1} + a_{2} b_{0} .

Only terms whose two exponents add to the target exponent contribute.

ii

Solution

In the Cauchy product,

a_{j} x^{j}

combines with

b_{i - j} x^{i - j}

to give a term of degree

i

. Hence

c_{i} = j = 0 \sum i a_{j} b_{i - j} .

This coefficient identity holds for formal power series. For convergent Taylor series it also gives the function product throughout the common open interval of convergence, where absolute convergence justifies regrouping.

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