Express the following limit as a definite integral. Show all the steps in your work.
Lim as n approaches infinity of (3/n * (n)sigma(I=1) * (1+(3i/n))/(9+(1+(3i/n)^2)).
I am utterly lost!
what is the formula for the ith term again? It seems garbled. There's a (3/n * floating around.
No worries! Let's go step by step to express this limit as a definite integral.
Step 1: Rewrite the limit using sigma notation.
The expression involves the summation of a function from i=1 to n, where n approaches infinity. However, to work with this limit more conveniently, we can rewrite it using the sigma notation. Here's the rewritten expression:
lim as n approaches infinity [ 3/n * Σ(1+(3i/n))/(9+(1+(3i/n)^2)) ] for i = 1 to n
Step 2: Calculate Δx, the width of each subinterval.
In this problem, Δx represents the width of each subinterval. Since we are summing from i=1 to n, we can express Δx as (b-a)/n, where a = 1 and b = n.
Step 3: Express the sum as an integral.
Now, we can transform the summation into an integral by multiplying the function by Δx and summing it from i=1 to n. Here's the expression after transforming it into an integral:
lim as n approaches infinity [ Σ(3 * Δx * (1+(3i/n))/(9+(1+(3i/n)^2))) ] for i = 1 to n
Step 4: Simplify the expression.
To simplify the expression further, we can substitute Δx, which we defined in step 2, into the integral. The expression now becomes:
lim as n approaches infinity [ Σ(3 * ((b-a)/n) * (1+(3i/n))/(9+(1+(3i/n)^2))) ] for i = 1 to n
Step 5: Convert the sum to an integral.
We can now convert the sum to an integral and change the limits of integration from i=1 to i=n to x=a to x=b. Here's the updated expression:
lim as n approaches infinity [ ∫(a to b) 3 * ((b-a)/n) * (1+(3x/n))/(9+(1+(3x/n)^2)) dx ]
Step 6: Simplify the expression further.
Lastly, we can simplify the expression by factoring out the constants and integrating the function with respect to x. Here's the final expression as a definite integral:
lim as n approaches infinity [ (b-a)/n ∫(a to b) (3/9) * (1+(3x/n))/(1+((3x/n)^2)) dx ]
And that's it! You have successfully expressed the given limit as a definite integral, showing all the steps along the way.