# Find all values of x in which the series converges

• turbokaz
In summary, a series converges when its sequence of partial sums approaches a finite number as the number of terms increases. To determine convergence, tests such as the comparison, ratio, and root tests can be used, although they do not guarantee convergence. Finding all values of x in which a series converges allows us to determine its domain of convergence, which can be useful in various applications. It is possible for a series to converge for some values of x and diverge for others, known as conditional convergence. However, there is no general method for finding all values of x in which a series converges, and it often requires a combination of tests and techniques.
turbokaz

## Homework Statement

Find all values of x for which the series Ʃ (4x+5)^n/2^n converges. (n=0; n→∞)
(3/4,7/4)
(-7/4,7/4)
(-5/4,5/4)
(-7/4,-3/4)

## The Attempt at a Solution

I think the answer is -7/4,-3/4. If I remember correctly, there is a test that states that a series converges if the limit goes to 0. Is this correct?

hi turbokaz!

look at it this way …

it's ∑ an where a = (4x+5)/2

so what is the equation for when ∑ an converges?

## 1. What does it mean for a series to converge?

When a series converges, it means that the sequence of its partial sums approaches a finite number as the number of terms in the series increases.

## 2. How can I determine if a series converges?

There are several tests that can be used to determine if a series converges, such as the comparison test, the ratio test, and the root test. It is important to note that these tests do not guarantee convergence, but rather provide a way to analyze the behavior of a series.

## 3. What is the significance of finding all values of x in which a series converges?

Finding all values of x in which a series converges allows us to determine the domain of convergence for the series. This can be useful in applications such as calculating the value of a function at a specific point or finding the area under a curve.

## 4. Can a series converge for some values of x and diverge for others?

Yes, it is possible for a series to converge for some values of x and diverge for others. This is known as conditional convergence and typically occurs when the series alternates between positive and negative terms.

## 5. Is there a general method for finding all values of x in which a series converges?

No, there is not a single method that can be used to find all values of x in which a series converges. It often requires a combination of different tests and techniques to determine the convergence of a series.

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