Problem 6

Let

a_1,\ldots,a_n

be positive numbers and let

0<q<1

. Find real numbers

b_1,\ldots,b_n

such that (1)

a_k<b_k

for every

k

; (2)

q<b_{k+1}/b_k<1/q

for

k=1,\ldots,n-1

; and (3)

\sum_{k=1}^n b_k<\frac{1+q}{1-q}\sum_{k=1}^n a_k

Step 3 of 4: Bound the total

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In plain words

The finite kernel sum is dominated by the infinite geometric kernel.

\sum_{r=1}^n b_r=\sum_{i=1}^n a_i\sum_{r=1}^n q^{|r-i|}

Detailed analysis

Swap the finite sums. For each $i$ , the inner sum is strictly smaller than the bilateral geometric sum $\sum_{j\in\mathbb Z}q^{|j|}=(1+q)/(1-q)$ , because the finite index interval omits some positive terms.