91Ó°ÊÓ

Finding Partial Sums In Exercises \(1-6,\) find the sequence of partial sums \(S_{1}, S_{2}, S_{3}, S_{4},\) and \(S_{5}\) . $$ 1+\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+\frac{1}{25}+\cdots $$

Using the Integral Test In Exercises \(1-22,\) confirm that the Integral Test can be applied to the series. Then use the Integral Test to determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty} \frac{1}{n+3} $$

In Exercises 1–6, write the first five terms of the sequence. $$ a_{n}=3^{n} $$

Finding the Center of a Power Series In Exercises \(1-4\) , state where the power series is centered. $$ \sum_{n=0}^{\infty} n x^{n} $$

In Exercises 1–4, find a geometric power series for the function, centered at 0, (a) by the technique shown in Examples 1 and 2 and (b) by long division. $$ f(x)=\frac{1}{4-x} $$

Finding a Taylor Series In Exercises \(1-12,\) use the definition of Taylor series to find the Taylor series, centered at \(c,\) for the function. $$ f(x)=e^{2 x}, \quad c=0 $$

Verify the formula. \(\frac{(n+1) !}{(n-2) !}=(n+1)(n)(n-1)\)

Using the Integral Test In Exercises \(1-22,\) confirm that the Integral Test can be applied to the series. Then use the Integral Test to determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty} \frac{2}{3 n+5} $$

Finding the Center of a Power Series In Exercises \(1-4\) , state where the power series is centered. $$ \sum_{n=1}^{\infty} \frac{(-1)^{n} 1 \cdot 3 \cdot \cdot \cdot(2 n-1)}{2^{n} n !} x^{n} $$

Verify the formula. \(\frac{(2 k-2) !}{(2 k) !}=\frac{1}{(2 k)(2 k-1)}\)