Series expansion

Expand $\sqrt{4-3x}$ through $x^2$, state its validity interval, and compare two approximations to $\sqrt{13}$.

Expand $\left(\dfrac{1-x}{1+2x}\right)^{1/3}$ through $x^2$, then approximate $\sqrt{5}$.

Decompose $\dfrac{2x^2+3x-1}{(x-1)(x^2+1)}$, expand through $x^4$, and find the coefficient of $x^{2008}$.

Compare the $x^3$ coefficient in $(2+ax)^{-5}$ with the $x^2$ coefficient in $\sqrt{4-ax}$ to find $a$.

Use the first three terms of $(2-ax)^{-b}$ to find $a,b,c$, then derive its $x^r$ coefficient.

Find the first three terms of $\dfrac{1+x^2}{\sqrt{3-2x}}$, its validity interval, and a tangent at $x=0$.

Decompose $\dfrac{9+4x^2}{(1-2x)^2(2+x)}$ and find both its first four terms and its $x^r$ coefficient.

Expand $\ln(1+e^{2x})$ through $x^2$, test a tolerance of $\pm0.3$, and deduce a related series.

Find the series of $y=\sqrt{e^x\cos x}$ through $x^2$ using derivatives and standard expansions.

Expand $e^{\sin3x}$ and use the series to approximate a definite integral; compare with a calculator value.

Expand $\ln(k+2x)^n$, then estimate an integral and discuss the percentage error.

Given $2yy'=y^2-1$ and $(0,3)$, find derivatives at zero and expand $y$ through $x^3$.

Derive a differential identity for $y=\sqrt{\cos x}$ and use its series for a numerical approximation.

Find derivatives of $\ln(2x+1)$ at zero, then expand $\ln(2x^2+3x+1)$.

Use a differential identity to expand $\cos[\ln(1+x)]$, then deduce the sine series and a tangent.

Expand $\ln(2+\sin2x)$ by two methods, then expand a quotient involving $\sqrt{1-x}$.

Use the small-angle series of $\tan\theta$ to approximate a length in triangle OBC.

Use the angles and side PR of triangle PQR to approximate PQ, then test a 5% error bound.

Given OP = 5 cm, OQ = 7 cm and angle $\theta$, approximate the perimeter containing a circular arc.

Expand $y=\sqrt{1+e^{\tan x}}$ and its derivative, then estimate $\int_{-1}^1 y\,dx$.

Differentiate $y=\sqrt2\tan^{-1}(\dfrac{x\sqrt2}{x+3})$ to build its series and a related rational series.

For $y=\cot(2x+\dfrac\pi4)$, derive a differential identity, expand through $x^3$ and estimate a cosecant value.

Given $y'=\dfrac{e^{\tan^{-1}x}}{1+x^2}$ and $y(0)=1$, find $y$, build its series and deduce two related expansions.