# What is the interval of convergence of #sum_1^oo (3x-2)^(n)/(1+n+n^(x) #?

The series converges for

Hence, the series will undoubtedly converge for:

that is intended for:

If we format this as follows:

Additionally, the first series is convergent, just as the second series is the alternate armonic series.

The integral test can be used to demonstrate the non-convergence of this series:

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

To find the interval of convergence of the series (\sum_{n=1}^{\infty} \frac{(3x-2)^n}{1+n+n^x}), we can use the ratio test. The ratio test states that a series (\sum_{n=1}^{\infty} a_n) converges absolutely if (\lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right| < 1).

Applying the ratio test to the given series:

[ \lim_{n \to \infty} \left| \frac{\frac{(3x-2)^{n+1}}{1+(n+1)+(n+1)^x}}{\frac{(3x-2)^n}{1+n+n^x}} \right| < 1 ]

After simplifying:

[ \lim_{n \to \infty} \left| \frac{(3x-2)^{n+1}(1+n+n^x)}{(3x-2)^n(1+(n+1)+(n+1)^x)} \right| < 1 ]

[ \lim_{n \to \infty} \left| \frac{(3x-2)(1+n+n^x)}{1+(n+1)+(n+1)^x} \right| < 1 ]

To find the interval of convergence, solve the inequality:

[ |3x - 2| \lim_{n \to \infty} \frac{1+n+n^x}{1+n+ (n+1)^x} < 1 ]

By considering the behavior of the limit as (n) approaches infinity, determine the range of (x) for which the inequality holds true.

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.

- How do you find the radius of convergence #Sigma (1*4*7* * * (3n+1))/(n!)x^n# from #n=[0,oo)#?
- How do you find a power series representation for #f(x)=ln(1+x)# and what is the radius of convergence?
- How do you find the Taylor polynomial of degree 10 of the function #arctan(x^3)# at a = 0?
- What is the Maclaurin series for? : #sqrt(1-x)#
- How do I construct a Taylor series for #f(x)=1/sqrt(x)# centered at x=4?

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