Use series to evaluate the following limit.

$$lim x→0 sin x-x/x^3$$

Evaluate the limit. $\lim _{x \rightarrow 0}(x \csc x)$.

Evaluate the following limits.

$$\lim _{x \rightarrow 0} x \csc ^{2} \sqrt{2 x}$$

Use an appropriate Taylor series to find the first four nonzero terms of an infinite series that is equal to the following numbers. $\ln \frac { 3 } { 2 }$

Find the first four nonzero terms of the Taylor series for the given function centered at a.

Find the first four nonzero terms of the Maclaurin series for the given function. f(x)=ln(1+4x)

Find the first four terms of the binomial series for the function.

$$( 1 - x ) ^ { - 3 }$$

Determine whether the series is convergent or divergent. If it is convergent, find its sum. summation n=1 to infinity [(0.8)^n-1-(0.3)^n]

Use the rectangle s in each graph to approximate the area of the reg ion bounded by y = 5 / x, y = 0, x = I, and x = 5. Describe how you could continue this process to obtain a more accurate approximation of the area.

Show that the function represented by the power series is a solution of the differential equation. y = Σ_(n=0)^∞ (-1)^n x^2n+1 / (2n+1)!, y″+y=0

A diving board 3.00 m long is supported at a point 1.00 m from the end, and a diver weighing 500 N stands at the free end. The diving board is of uniform cross section and weighs 280 N. Find the force at the support point.

Find the first 40 terms of the sequence defined by a n+1 ={1/2an if an is an even number, 3 an + 1 if an is an odd number. and a1 = 11. Do the same if a1 = 25. Make a conjecture about this type of sequence.

Is there a number a such that lim x tends to -2 3x^2+ax+a+3/x^2+x-2 exists? If so, find the value of a and the value of the limit.

Find the distance traveled in 15 seconds by an object moving with a velocity of v(t) = 20 + 7 cos t feet per second.