How do you find a formula for the sum n terms #Sigma1/n^3(i-1)^2# and then find the limit as #n->oo#?
# sum_(i=1)^n 1/n^3(i-1)^2 = 1/(6n^2)(2n-1)(n-1) #
# lim_(n rarr oo)sum_(i=1)^n 1/n^3(i-1)^2 = 1/3 #
We have;
By signing up, you agree to our Terms of Service and Privacy Policy
To find a formula for the sum of n terms of the series ( \sum_{i=1}^{n} \frac{1}{n^3(i-1)^2} ), we can first rewrite the series in a telescoping form. Factoring out ( \frac{1}{n^3} ) from each term yields:
[ \frac{1}{n^3} \sum_{i=1}^{n} \frac{1}{(i-1)^2} ]
Next, we can use the formula for the sum of squares of consecutive integers:
[ \sum_{i=1}^{n} i^2 = \frac{n(n + 1)(2n + 1)}{6} ]
By substituting ( i - 1 ) for ( i ), we get:
[ \sum_{i=1}^{n} (i-1)^2 = \frac{n(n - 1)(2n - 1)}{6} ]
So, the sum of the series becomes:
[ \frac{1}{n^3} \cdot \frac{n(n - 1)(2n - 1)}{6} ]
[ = \frac{(n - 1)(2n - 1)}{6n^2} ]
To find the limit of this expression as ( n ) approaches infinity, we can apply the limit properties. After simplifying the expression, we get:
[ \lim_{n \to \infty} \frac{(n - 1)(2n - 1)}{6n^2} = \lim_{n \to \infty} \frac{2n^2 - 3n + 1}{6n^2} ]
[ = \lim_{n \to \infty} \left( \frac{2}{6} - \frac{3}{6n} + \frac{1}{6n^2} \right) ]
[ = \frac{1}{3} ]
Therefore, the limit of the series ( \sum_{i=1}^{n} \frac{1}{n^3(i-1)^2} ) as ( n ) approaches infinity is ( \frac{1}{3} ).
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.

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