How do you test the series #Sigma n/((n+1)(n^2+1))# from n is #[0,oo)# for convergence?
To test the series ( \sum \frac{n}{(n+1)(n^2+1)} ) for convergence, you can use the ratio test.
-
Apply the ratio test by computing the limit of the absolute value of the ratio of successive terms as ( n ) approaches infinity:
[ \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| ]
where ( a_n = \frac{n}{(n+1)(n^2+1)} ).
-
Compute ( a_{n+1} ) and ( a_n ) separately:
[ a_{n+1} = \frac{n+1}{(n+2)((n+1)^2+1)} ] [ a_n = \frac{n}{(n+1)(n^2+1)} ]
-
Take the absolute value of the ratio of successive terms:
[ \left| \frac{a_{n+1}}{a_n} \right| = \frac{\frac{n+1}{(n+2)((n+1)^2+1)}}{\frac{n}{(n+1)(n^2+1)}} ]
-
Simplify the expression and compute the limit as ( n ) approaches infinity.
-
If the limit is less than 1, the series converges. If the limit is greater than 1 or the limit does not exist, the series diverges.
By following these steps, you can determine the convergence or divergence of the given series.
By signing up, you agree to our Terms of Service and Privacy Policy
The series:
is convergent.
As the series:
is convergent based on the p-series test, then also the series:
is convergent.
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.
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 apply the ratio test to determine if #Sigma 1/sqrtn# from #n=[1,oo)# is convergent to divergent?
- Does the Alternating Series Test determine absolute convergence?
- How do you find #\lim _ { x \rightarrow \infty } \sqrt { x ^ { 2} + 1x - 4} - x#?
- How do you find #a_1# for the geometric series with #r=3# and #s_6=364#?
- How do you find #lim cos(3theta)/(pi/2-theta)# as #theta->pi/2# using l'Hospital's Rule?

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