How you solve this?: #lim_(n->oo)sum_(k=1)^n(2k+1)/(k^2(k+1)^2)#

Answer 1

#1#

We know that

#(2 k + 1)/(k^2 (k + 1)^2)=1/k^2-1/(k+1)^2# then
#sum_(k=1)^n(2 k + 1)/(k^2 (k + 1)^2)=sum_(k=1)^n 1/k^2-sum_(k=1)^n 1/(k+1)^2 = 1-1/(n+1)^2# so
#lim_(n->oo)sum_(k=1)^n(2 k + 1)/(k^2 (k + 1)^2)=1#
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Answer 2

To solve the limit as n approaches infinity of the sum from k equals 1 to n of (2k+1) divided by (k squared multiplied by (k+1) squared), we can use the limit comparison test.

First, let's simplify the expression inside the sum. We can rewrite (2k+1) as 2k+1 = 2k+2-1 = 2(k+1)-1.

Now, let's rewrite the sum using this simplified expression:

lim_(n->oo) sum_(k=1)^n (2k+1)/(k^2(k+1)^2) = lim_(n->oo) sum_(k=1)^n (2(k+1)-1)/(k^2(k+1)^2)

Next, we can split the sum into two separate sums:

lim_(n->oo) sum_(k=1)^n (2(k+1)-1)/(k^2(k+1)^2) = lim_(n->oo) sum_(k=1)^n (2(k+1))/(k^2(k+1)^2) - lim_(n->oo) sum_(k=1)^n (1)/(k^2(k+1)^2)

Now, let's evaluate each sum separately.

For the first sum, we can simplify the expression inside the sum by canceling out the (k+1) terms:

lim_(n->oo) sum_(k=1)^n (2(k+1))/(k^2(k+1)^2) = lim_(n->oo) sum_(k=1)^n 2/(k^2(k+1))

For the second sum, we can simplify the expression inside the sum by canceling out the (k+1) terms:

lim_(n->oo) sum_(k=1)^n (1)/(k^2(k+1)^2) = lim_(n->oo) sum_(k=1)^n 1/(k^2(k+1))

Now, let's evaluate each of these sums using the limit comparison test.

For the first sum, we can compare it to the sum 2/(k^3) since the highest power of k in the denominator is 3.

lim_(n->oo) sum_(k=1)^n 2/(k^2(k+1)) = lim_(n->oo) sum_(k=1)^n 2/(k^3)

For the second sum, we can compare it to the sum 1/(k^3) since the highest power of k in the denominator is 3.

lim_(n->oo) sum_(k=1)^n 1/(k^2(k+1)) = lim_(n->oo) sum_(k=1)^n 1/(k^3)

Both of these new sums are known convergent p-series with p=3. Therefore, both sums converge.

By the limit comparison test, since both sums converge, the original sum also converges.

Hence, the solution to the given limit is that it converges.

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Answer from HIX Tutor

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.

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