# How do you use the summation formulas to rewrite the expression #Sigma (2i+1)/n^2# as i=1 to n without the summation notation and then use the result to find the sum for n=10, 100, 1000, and 10000?

Let

# S_n = sum_(i=1)^n (2i+1)/n^2 #

# :. S_n = 1/n^2sum_(i=1)^n (2i+1) #

# :. S_n = 1/n^2{2sum_(i=1)^n (i)+sum_(i=1)^n (1)} #

And using the standard results:

We have;

# \ \ \ \ \ S_n = 1/n^2{2*1/2n(n+1) + n} #

# :. S_n = 1/n^2{n(n+1) + n} #

# :. S_n = 1/n^2{n^2+n + n} #

# :. S_n = 1/n^2{n^2+2n} #

# :. S_n = 1/n^2{n(n+2)} #

# :. S_n = (n+2)/n #

And this has been calculated using Excel for

What happens as

[ NB As an additional task we could possibly conclude that as

Now,

# :. S_n = 1+2/n #

And so,

# lim_(n rarr oo) S_n = lim_(n rarr oo) (1+2/n) #

# :. lim_(n rarr oo) S_n = lim_(n rarr oo) (1) + 2lim_(n rarr oo)(1/n) #

# :. lim_(n rarr oo) S_n = 1+0 #

# :. lim_(n rarr oo) S_n = 1#

Which confirms our assumption!

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

To rewrite the expression ( \sum_{i=1}^{n} \frac{2i+1}{n^2} ) without summation notation, you can use the property of linearity of summation. Then, you can find the sum by plugging in the values of ( n ) as 10, 100, 1000, and 10000 into the resulting expression.

The expression can be rewritten as:

[ \frac{2}{n^2} \sum_{i=1}^{n} i + \frac{1}{n^2} \sum_{i=1}^{n} 1 ]

[ = \frac{2}{n^2} \left( \frac{n(n+1)}{2} \right) + \frac{1}{n^2} \left( \sum_{i=1}^{n} 1 \right) ]

[ = \frac{n+1}{n} + \frac{1}{n^2} \left( n \right) ]

[ = \frac{n+1}{n} + \frac{1}{n} ]

[ = 1 + \frac{1}{n} ]

Using this expression, we can find the sum for ( n = 10, 100, 1000, ) and ( 10000 ):

For ( n = 10 ): [ = 1 + \frac{1}{10} = 1.1 ]

For ( n = 100 ): [ = 1 + \frac{1}{100} = 1.01 ]

For ( n = 1000 ): [ = 1 + \frac{1}{1000} = 1.001 ]

For ( n = 10000 ): [ = 1 + \frac{1}{10000} = 1.0001 ]

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 integrate #int y^2sqrtydy#?
- How do you find the Improper integral #int (x^2)e^[(-x^2)/2] dx # from x=-∞ to x=∞?
- How do you find the indefinite integral of #int (12/x^4+8/x^5) dx#?
- What is the net area between #f(x) = sqrt(x^2+2x+1) # and the x-axis over #x in [2, 4 ]#?
- How do you find antiderivative of #(1-x)^2#?

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