Suppose that the Federal Reserve (the Fed) buys $100 million of government debt obligations from private owners. This creates$100 million of new money and sets off a chain reaction because of the "fractional reserve" banking system. When the $100 million is deposited into private bank accounts, the banks keep only 15$(.85)^2(100)$million additional dollars. This process may be repeated indefinitely. Compute the total amount of new money that can be created theoretically by this process, beyond the original$100 million. (In practice, only about an additional $300 million is created, usually within a few weeks of the action of the Fed.)

Determine the sums of the following infinite series:

$$\sum_{k=1}^{\infty}\left(\frac{1}{3}\right)^{2 k}$$

In Exercise given below, find an infinite series that converges to the value of the given definite integral.

$$\int_0^1 x e^{x^3} d x$$

In Exercise given below, use three repetitions of the Newton-Raphson algorithm to approximate the following:

$$\sqrt[3]{11}$$

In Exercise given below, determine the third Taylor polynomial of the given function at x=0.

$$f(x)=\cos (\pi-5 x)$$

In Exercise given below, find the Taylor series at x=0 of the given function by computing three or four derivatives and using the definition of the Taylor series.

$$(1+x)^3$$

Find the nth Taylor polynomial of $\frac{2}{2-x}$ at x=0.

Determine the sums of the following geometric series when they are convergent.

$$1+\frac{1}{2^3}+\frac{1}{2^6}+\frac{1}{2^9}+\frac{1}{2^{12}}+\cdots$$

In Exercise given below, use the integral test to determine whether the infinite series is convergent or divergent. (You may assume that the hypotheses of the integral test are satisfied.)

$$\sum_{k=0}^{\infty} \frac{7}{k+100}$$

Explain how the Newton-Raphson algorithm is used to approximate a zero of a function.

Let $f(x)=1+x^2+x^4+x^6+\cdots$.

(a) Find the Taylor series expansion of $f^{\prime}(x)$ at x=0.

(b) Find the simple formula for $f^{\prime}(x)$ not involving a series. [Hint: First find a simple formula for f(x).]

In Exercise given below, find an infinite series that converges to the value of the given definite integral.

$$\int_0^1 e^{-x^2} d x$$

Determine the sums of the following infinite series:

$$\sum_{k=0}^{\infty}(-1)^k \frac{3^{k+1}}{5^k}$$

Let $f(x)=\ln |\sec x+\tan x|$. It can be shown that $f^{\prime}(0)=1, f^{\prime \prime}(0)=0, f^{\prime \prime \prime}(0)=1$, and $f^{(4)}(0)=0$. What is the fourth Taylor polynomial of f(x) at x=0?