How do you test the series #Sigma (3n^2+1)/(2n^41)# from n is #[1,oo)# for convergence?
To test the convergence of the series ( \sum_{n=1}^\infty \frac{3n^2+1}{2n^41} ), you can use the ratio test.
Apply the ratio test:

Calculate the limit as ( n ) approaches infinity of the absolute value of the ratio of successive terms: [ L = \lim_{n \to \infty} \left \frac{a_{n+1}}{a_n} \right ] where ( a_n = \frac{3n^2+1}{2n^41} ).

Evaluate the limit: [ L = \lim_{n \to \infty} \left \frac{\frac{3(n+1)^2+1}{2(n+1)^41}}{\frac{3n^2+1}{2n^41}} \right ]

Simplify and compute the limit ( L ).

If ( L < 1 ), the series converges absolutely. If ( L > 1 ), the series diverges. If ( L = 1 ), the test is inconclusive.
This procedure will determine the convergence or divergence of the given series.
By signing up, you agree to our Terms of Service and Privacy Policy
is convergent based on the direct comparison test.
We can test the series by direct comparison. As:
Now we have:
is convergent
By signing up, you agree to our Terms of Service and Privacy Policy
When evaluating a onesided 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 onesided 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 onesided 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 onesided 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 test the improper integral #int x^(3/2) dx# from #[0, oo)# and evaluate if possible?
 How do you determine the convergence or divergence of #sum_(n=1)^(oo) (1)^(n+1)/n#?
 How do you find the limit of #s(n)=64/n^3[(n(n+1)(2n+1))/6]# as #n>oo#?
 How do you use the limit comparison test to determine if #Sigma tan(1/n)# from #[1,oo)# is convergent or divergent?
 What is the sum of the series #1+ln2+(((ln2)^2)/(2!))+...+(((ln2)^n)/(n!))+...#?
 98% accuracy study help
 Covers math, physics, chemistry, biology, and more
 Stepbystep, indepth guides
 Readily available 24/7