# How do you determine if #a_n=1+3/7+9/49+...+(3/7)^n+...# converge and find the sums when they exist?

Convergent.

Sum to infinity

This is a geometric series:

Where:

If:

So we have:

The sum of a geometric series is given by:

From this we can see that if:

This is a finite value, and the sum is said to converge.

If:

The sum increases without bound, and is said to diverge.

By signing up, you agree to our Terms of Service and Privacy Policy

This is a geometric series with first term ( a_0 = 1 ) and common ratio ( r = \frac{3}{7} ). The series converges if ( |r| < 1 ). In this case, ( |3/7| < 1 ) so the series converges.

The sum of a convergent geometric series is given by the formula ( S = \frac{a}{1 - r} ), where ( a ) is the first term and ( r ) is the common ratio.

Substituting the values, we get ( S = \frac{1}{1 - \frac{3}{7}} = \frac{1}{\frac{4}{7}} = \frac{7}{4} ). Thus, the sum of the series is ( \frac{7}{4} ).

By signing up, you agree to our Terms of Service and Privacy Policy

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- For the sequence 1/3, 1/3^2 ,1/3^3 ,1/3^4 ,1/3^5,…, ?
- How do you determine if the improper integral converges or diverges #int (1/(3x)-6) dx# from negative infinity to 0?
- How do you use the Ratio Test on the series #sum_(n=1)^oo(-10)^n/(4^(2n+1)(n+1))# ?
- Find the limit of the sequence #ln(n^2+2) -1/2ln(n^4+4)#. Does it converge to 0?
- How do you test the alternating series #Sigma (-1)^(n+1)/sqrtn# from n is #[1,oo)# for convergence?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7