Chapter 8: Infinite Sequences and Series
Section 8.2: Series
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?
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).
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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