In \(\sum_{n=0}^{\infty}ax^n\), what is the common ratio?

State the condition for \(\sum_{n=0}^{\infty}ax^n\) to converge when \(a\ne0\).

Find the first four terms of \(\sum_{n=0}^{\infty}5(1/3)^n\).

For \(4+2+1+1/2+\cdots\), identify \(a\) and \(x\).

Compute \(\sum_{n=0}^{\infty}3(1/2)^n\).

Compute the finite sum \(\sum_{n=0}^{3}2^n\).

Use the finite geometric formula to find \(\sum_{n=0}^{4}3(2)^n\).

Decide whether \(\sum_{n=0}^{\infty}7(-0.4)^n\) converges, and find its sum if it does.

Find \(\sum_{n=1}^{\infty}2(1/3)^n\).

Derive \(S_N=a(1-x^N)/(1-x)\) for \(S_N=a+ax+\cdots+ax^{N-1}\), with \(x\ne1\).

Explain why the formula \(a/(1-x)\) is invalid for \(\sum_{n=0}^{\infty}a(1.2)^n\).

A reflected pulse keeps \(60\%\) of its previous size. If the first contribution is \(10\), find the total contribution of infinitely many reflections.

For what real values of \(r\) does \(\sum_{n=0}^{\infty}4r^n\) converge?

Find \(r\) if \(\sum_{n=0}^{\infty}6r^n=15\) and \(|r|<1\).

Find the smallest \(N\) such that the remainder after \(N\) terms of \(\sum_{n=0}^{\infty}(1/2)^n\) is less than \(1/16\).

For which real \(x\) does \(\sum_{n=0}^{\infty}(x-2)^n\) converge?

Compute \(\sum_{n=2}^{\infty}5(1/4)^n\) carefully accounting for the starting index.

Diagnose the error: \(\sum_{n=0}^{\infty}2^n=1/(1-2)=-1\).

Show from partial sums why \(\sum_{n=0}^{\infty}a\) diverges when \(a\ne0\).

A finite geometric sum is \(S_N=12(1-0.7^N)\). Identify the infinite sum and explain why it exists.