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Calculus 2

Determine the radius of convergence for the power series with cn=n!c_n = n! and (x2)n(x - 2)^n.

Express 11+x2\frac{1}{1 + x^2} as the sum of a power series.

Find the interval and radius of convergence for the power series representation of 11+x2\displaystyle \frac{1}{1+x^2}.

Find a power series representation for f(x)f(x) and determine the interval of convergence.

Express the function as a sum of a power series by first using partial fractions, then determine the interval of convergence.

Determine the interval of convergence for the series n=0(1)nxn4n+1\sum_{n=0}^{\infty} \frac{(-1)^n x^n}{4^{n+1}}.

Use the root test to determine if the series from 1 to infinity of 14n\frac{1}{4^n} will converge or diverge.

Use the root test to determine if the series (3+5n2+3n)n\left(\frac{3+5n}{2+3n}\right)^n converges, diverges, or is inconclusive.

Apply the root test to the series (1n2+1n)n\left(\frac{1}{n^2 + \frac{1}{n}}\right)^n.

Use the root test on the series (nn21+4n)\left(\frac{n^n}{2^{1+4n}}\right) to determine its convergence or divergence.

Determine if the series (n2n)\displaystyle \left(\frac{n}{2^n}\right) converges or diverges using the root test.

Apply the root test to the series (1)n3n+2(n1)n(-1)^n \cdot \frac{3^{n+2}}{(n-1)^n} to determine if it converges or diverges.

Using the Ratio Test or the Root Test, determine if the infinite series an\sum a_n converges or diverges.

Apply the root test to check if a series converges absolutely or diverges.

Using the ratio or root test, determine if the series [{ "type" : "katexPlugin", "text": "\sum a_n", "inline": true }]', is absolutely convergent.

Look at the Taylor polynomial for cos(x)\cos(x) and we're cutting it off at degree 4, T4(x)T_4(x). We want to figure out what values of xx you can plug in there for it to be accurate to two decimal places.

Using a Taylor polynomial, approximate a function when x is in the range [7,upper bound][7, \, \text{upper bound}].

Find the Maclaurin series for the function f(x)=exf(x) = e^x.

Find the Taylor series for the function f(x)=exf(x) = e^x centered at a=2a = 2.