Explanation:

This is a geometric series. Use the adhering to formula, where

is the first term of the series, and
is the ratio that need to be much less than 1. If
is better than 1, the series diverges.

You watching: Questions about series calculus

Rationalize the denominator.

Consider the following summation:

. Does this converge or diverge? If it converges, wright here does it approach?

Explanation:

The difficulty can be reconverted utilizing a summation symbol, and also it deserve to be checked out that this is geometric.

Because the ratio is much less than 1, this series will certainly converge. The formula for geometric series is:

where

is the first term, and
is the widespread proportion. Substitute these worths and also resolve.

A worm crawls up a wall surface during the day and slides down slowly during the night. The first day the worm crawls one meter up the wall. The first night the worm slides down a third of a meter. The second day the worm regains one third of the lost development and slides dvery own one third of that distance reacquired on the second night. This pattern of motion continues...

Which of the adhering to is a geometric sum representing the distance the worm has actually travelled after

12-hour periods of motion? (Assuming day and also night are both 12 hour periods).

Explanation:

The sum need to be alternating, and also after one duration you need to have the worm at 1m. After 2 periods, the worm need to be at 2/3m. There is just one sum for which that is true.

Explanation:

By writing out the initially few terms and factoring out a 10, we deserve to arrive at a correct representation of the amount in +... notation.

The amount of the initially term wright here k=4 is

.

The term once k=5 is

.

The term as soon as k=6 is

.

Creating the summation we get,

One approach for determining whether a series converges is comparing it to a related imcorrect integral.

Explanation:

In the spirit of the direct comparichild test, we deserve to compare the convergent improper integral to the unlimited amount, such that the amount is less than the convergent integral by interpreting the sum as a right hand also Riemann amount. Note that you have the right to depict the sum as either, through the property that in one situation you get information about its convergence or divergence and in the various other case you cannot say anything. We desire to make certain for correctness that we assert the sum and also integral are both positive, as a condition for making the compariboy.

By meaning of the Comparison Test

Given that

where
.

If

converges, then
also converges.

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### Example Question #3 : Series And Functions

To what value does the sum

converge?

Explanation:

The first term of the series is

, and also the widespread proportion is
, so we have the right to recompose the series in the form
and also usage the formula
to arrive at the correct answer.

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### Example Inquiry #4 : Series And Functions

Given that

converges, what can you say about
where
and
.

The sum need to converge.

Not enough information.

The amount is zero.

The amount must diverge.

The amount should converge.

Explanation:

The amount have to converge by the direct comparikid test. All terms are positive and

is smaller sized than
for all k. Because of this by meaning the sum must converge.

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### Example Question #5 : Series And Functions

A worm crawls up a wall during the day and slides dvery own progressively during the night. The initially day the worm crawls one meter up the wall. The initially night the worm slides down a 3rd of a meter. The second day the worm regains one 3rd of the shed development and slides down one 3rd of that distance regained on the second night. This pattern of movement proceeds...

Which of the complying with is a correct expression for the distance the worm has actually took a trip after the second night?

Explanation:

This is simply a finite sum of the distance the worm travelled and also dropped each day.

The first day the worm traveled a postive one meter.

Over the night he lost one 3rd of a meter. This can be analyzed right into math terms as

.

He continued this proportion development each day.

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### Example Concern #6 : Series And Functions

A worm crawls up a wall during the day and also slides down progressively during the night. The initially day the worm crawls one meter up the wall. The initially night the worm slides dvery own a third of a meter. The second day the worm regains one 3rd of the shed development and also slides down one 3rd of that distance reacquired on the second night. This pattern of activity proceeds...

How high off the ground will the worm eventually end up if he keeps at it forever?

Explanation:

This is a geometric series with an

and
so will converge to:

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### Example Concern #7 : Series And Functions

Which is the correct Maclaurin series depiction for

?

Explanation:

The general develop for the Maclaurin series for

is

To uncover the series representation for

, ssuggest substitute
in place of
in the series for
.

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