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**Unformatted text preview: **move is to ‘kick out’ the terms which we cannot
combine and rewrite the summations so that we can combine them. To that end, we note
k
j =0 k k+1−j j
a
b = ak+1 +
j k
j =1 k k+1−j j
a
b
j and
k
j =0 k k−j j +1
ab
=
j k −1 k k−j j +1
ab
+ bk+1
j j =0 so that
k k k+1−j j
a
b+
j (a + b)k+1 = ak+1 +
j =1 k −1
j =0 k k−j j +1
ab
+ bk+1
j We now wish to write
k
j =1 k k+1−j j
a
b+
j k−1
j =0 k k−j j +1
ab
j as a single summation. The wrinkle is that the ﬁrst summation starts with j = 1, while the second
starts with j = 0. Even though the sums produce terms with the same powers of a and b, they do
so for diﬀerent values of j . To resolve this, we need to shift the index on the second summation so
that the index j starts at j = 1 instead of j = 0 and we make use of Theorem 9.1 in the process.
k−1
j =0 k k−j j +1
ab
=
j k−1+1 k
ak−(j −1) b(j −1)+1
j−1 j =0+1
k =
j =1 k
ak+1−j bj
j−1 We can now combine our two sums using Theorem 9.1 and simplify using Theorem 9.3
k
j =1 k k+1−j j
a
b+
j k−1
j =0 k k−j j +1
ab
=
j k
j =1
k =
j =1
k =
j =1 Using...

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