Examples

Geometric series: algebra meets calculus

Consider the geometric series as a power series functions:

Take the derivative of both sides of the function:

This means satisfies the identity:

Now compute the derivative of the series:

On the other hand, compute the square of the series:

So we find that the same relationship holds, namely , for the closed formula and the series formula for this function.

Manipulating geometric series: algebra

Find power series that represent the following functions:

(a) (b) (c) (d)

Solution

(a)

Rewrite in format :

Choose . Plug into geometric series:

Therefore:

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(b)

Rewrite in format :

Choose . Plug into geometric series:

Therefore:

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(c)

Rewrite in format :

Choose . Here . Plug into geometric series:

Therefore:

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(d)

Rewrite in format :

Choose . Here . Plug into geometric series:

Therefore:

Manipulating geometric series: calculus

Find a power series that represents .

Solution

Differentiate to obtain similarity to geometric sum formula:

Integrate series to find original function:

Use known point to solve for :

Recognizing and manipulating geometric series: Part I

(a) Evaluate . (Hint: consider the series of .)

(b) Find a series approximation for .

Solution

(a)

(1) Follow hint, study series of :

Notice:

Integrate the series:

Solve for using which (plugging above) implies and thus . So:

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(2) Relate to the given series:

Notice that if we set . Also, . Therefore:

So the answer is .

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(b) Find a series approximation for :

Observe that .

Plug into the series:

Recognizing and manipulating geometric series: Part II

(a) Find a series representing using differentiation.

(b) Find a series representing .

Solution

(a)

Notice that .

What is the series for ?

Let :

Now integrate this by terms:

Conclude:

Plug in to solve for :

Final answer:

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(b)

Rewrite integrand in format of geometric series sum:

Therefore:

Integrate the series by terms to obtain the answer: