Chapter 8: Infinite Sequences and Series
Section 8.2: Series
Example 8.2.13
The Cauchy product of ∑n=0∞an and ∑n=0∞bn is the series ∑n=0∞cn, where cn=∑k=0nak⋅bn−k.
What happens to cn when the index in both the series being multiplied starts not at n=0, but n=1?
Solution
Table 8.2.13(a) lists cn,n=0,…,5, the first six members of the Cauchy product. From the listing in the table, generalize that if a0=b0=0, then for each ck, the first and last members are zero, and the terms of the Cauchy product reduce to the pattern shown in Table 8.2.13(b).
c0=a0b0
c0=0
Table 8.2.13(a) Terms of the Cauchy product
Table 8.2.13(b) Cauchy product when a0=b0=0
In practice, it appears to be easier to define a0 and b0 to be zero, and to use the existing definition of cn.
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