Question

Question #73f2e

Answer 1

What is the inequality to prove? I can't answer without it!

Explanation:

[What is the arithmetic-geometric sequence?]
An arithmetic-geometric sequence is the product of an arithmetic sequence and geometric sequence.

For example, a sequence
`1/1, 3/2. 5/4. 7/8, 9/16,…`
is an arithmetic-geometric sequence with general term `a_n=(2n-1)*(1/2)^(n-1)`.

More information:
https://en.wikipedia.org/wiki/Arithmetico%E2%80%93geometric_sequence

[Sample question]
Let `a_n=(2n-1)/2^(n-1)`. Prove that `lim_(n->oo) a_n=0`.

[How to solve?]
Generally, you can use the mathematical induction and the squeeze theorem to solve this type of inequation or limitation.

For example, if you can prove `0<2n-1<(3/2)^(n-1)` for `n>=8` with the mathematical induction, the inequation
`0<(2n-1)/2^(n-1)<(3/4)^(n-1)` is obtained.
Since `lim_(n->oo) (3/4)^(n-1)=0`, the sequence `a_n=(2n-1)/2^(n-1)` also converges to zero.

Alternatively, you can caluculate the sum `S_n=sum_(k=1)^n a_n` (the method is in the wikipedia page) and evaluate `lim_(n->oo)S_n`. If `S_n` converges, `a_n` must converge to zero.