What does \(R_N(x)\) measure?

State the alternating-series error estimate.

State the Taylor remainder bound with derivative bound \(M\).

What is the geometric tail after keeping terms through \(r^3\) in \(1+r+r^2+\cdots\)?

Using \(e^x\approx 1+x\), bound the error at \(x=0.05\) if \(e^t\le2\).

An alternating series has next omitted term size \(0.004\). What error bound follows?

Find the tail after keeping terms through \(n=2\) in \(\sum (1/4)^n\).

Use the alternating estimate to bound the error in \(\sin x\approx x-x^3/6\) at \(x=0.1\).

With \(e^t\le2\), test whether linear or quadratic terms of \(e^x\) are enough for error below \(0.001\) at \(x=0.1\).

Find the geometric tail after \(r^4\) when \(r=0.2\).

Why is a remainder estimate better than saying higher powers are small?

Why must the conditions of an error estimate be checked?

Bound the error in \(\cos x\approx1-x^2/2\) using the next even term for \(|x|\le0.2\).

Find the tail of \(\sum 3(0.5)^n\) after keeping through \(n=3\).

If \(|R_N|\le \frac{4(0.1)^{N+1}}{(N+1)!}\), test \(N=1\) and \(N=2\) against tolerance \(0.001\).

For \(\ln(1+x)=x-x^2/2+x^3/3-\cdots\), bound the error after \(x^3/3\) at \(x=0.1\).

If the first non-zero omitted term after degree \(2\) is degree \(5\), what happened to degrees \(3\) and \(4\)?

Diagnose the error: applying the alternating estimate to \(1+1/2+1/3+\cdots\).

Why can a conservative error bound still be useful?

Describe a strategy for checking whether a truncated series is accurate enough.