The sum of the first n terms of a series if given by the expression 6-2n+1/3n-1. By finding an expression for the nth

\text { Consider a sequence } f_{n}:[0, \pi] \rightarrow \mathbb{R} \text {, } f_{n}(x)=\left\{\begin{array}{ll} \sin (n x) & 0 \leq x \leq \frac{\pi}{n}, \\ 0 & \frac{\pi}{n} \leq x \leq \pi . \end{array}\right. ) Find pointwise limit of the sequence (fn) on [0, ]. Explain, why fn does not converge uniformly on1 [0, 1].

5: Decide whether the series converges. If so, find its sum. \sum_{n=1}^{\infty} \frac{3}{n(n+1)} \sum_{n=2}^{\infty} \frac{3^{n}-1}{5^{n-2}} Use the Ratio or Root test, to determine whether the series converges or diverges. \sum_{n=1}^{\infty} \frac{(\ln n)^{2 n}}{n^{n+1}}

A company estimates that it will need $107,000 in 14 years to replace a computer. It it establishes a sinking fund

\text { (7) Prove that the series } \sum_{k=1}^{\infty} \frac{\sin k}{k} \text { is conditionally convergent. }

5. (16 points) Determine whether the series listed below are divergent, absolutely convergent (hence convergent), or conditionally convergent. Indicate the tests or result you apply to support your conclusion. \text { a. } \sum_{n=1}^{\infty} \frac{(-1)^{n-1} e^{1 / n}}{n} \text { b. } \sum_{n=1}^{\infty} \frac{1}{n+n \tan ^{-1} n} \text { c. } \sum_{n=1}^{\infty} \frac{n !}{\ln n} \text { d. } \sum_{n=1}^{\infty} \frac{(-n)^{3}+1}{7^{n}}

In a large slab of thickness l the temperature p(x,t) at a distance x from one face satisfies the partial differential equation \frac{\partial \varphi}{\partial t}=\kappa \frac{\partial^{2} \varphi}{\partial x^{2}} where k is a positive constant. The faces x = 0 and x =l are both insulated (so that dp/dx = 0 on the faces) and at t = 0 the temperature for 0 < x < l is pox/l,where po is a constant. Use the method of separation of variables to show that y takes the form \varphi(x, t)=\frac{A_{0}}{2}+\sum_{n=1}^{\infty} A_{n} \cos \left(\frac{n \pi x}{l}\right) e^{-m^{2} \pi^{2} t / t^{2}} Determine the coefficients Ao and An (n= 1, 2,.). Write down the value of lim p(x, t) and give a brief physical interpretation.

Show that \sum_{k=0}^{\infty} \frac{(-1)^{k}}{2^{k}}=\frac{2}{3} (Hint. You could write the partial sums as a difference A – B where A contains the coefficients for even k, and B is the negative of the sum of the coefficients for odd k. Then consider A and B separatedly; geometric series.)

Find the equation for any horizontal asymptotes for the function below. f(x)=\frac{x^{2}+2 x+5}{x-20} Find the horizontal asymptote(s). Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.

1. Find explicit formulas for sequences of the form a₁, a2, a3, ... with the initial terms given below: 1/5, 3/20, 5/80, 7/320, 9/1280