f(x) = summation from n = 1 to ∞ of (x^n)((n + 1) / 4^(n +1)) with an interval of convergence, –4 < x < 4.
Find exactly the value of the integral from 0 to 2 of f(x), dx. Your answer will be a positive integer.
ALSO: Can you please take me through every step until you reach the ANSWER because I'm having trouble understanding this?
Thank you!
This article should prove helpful in relating the sum to the integral.
https://www.emathhelp.net/calculators/calculus-2/series-calculator/?f=%28n%2B1%29%2F4%5E%28n%2B1%29+x%5En&var=n&a=1&b=inf
To find the value of the integral from 0 to 2 of f(x), dx, we need to first evaluate the indefinite integral of f(x). Then, we can substitute the limits of integration (0 and 2) into that antiderivative expression and get the answer by subtracting the two results.
Step 1: Calculate the indefinite integral of f(x).
To integrate f(x), we need to find an antiderivative for each term in the summation individually. Let's start with the general term:
f(x) = ∑[(x^n)((n + 1) / 4^(n +1))]
Since the summation is from n = 1 to ∞, we can rewrite the expression to exclude the n = 0 term:
f(x) = ∑[(x^n)((n + 1) / 4^(n +1))]
= ∑[(x^n)(1 / 4^(n +1))] - [(x^0)(1 / 4^(0 + 1))]
= ∑[(x^n)(1 / 4^(n +1))] - 1/4
Now, we can perform the integration term by term:
∫[(x^n)(1 / 4^(n +1))] dx = [(x^(n+1))/(n+1)] * [1 / (4^(n+1) * (n+1))]
Step 2: Evaluate the indefinite integral from 0 to 2.
Substitute the limits of integration (0 and 2) into the expression we obtained in Step 1:
I = [(2^(n+1))/(n+1)] * [1 / (4^(n+1) * (n+1))] - [(0^(n+1))/(n+1)] * [1 / (4^(n+1) * (n+1))]
To evaluate the integral, we substitute 2 and 0 for x and simplify:
I = [(2^(n+1))/(n+1)] * [1 / (4^(n+1) * (n+1))]
Step 3: Simplify the expression and find the definite integral.
Since we want to find the value of the integral from 0 to 2, we need to subtract the result when x = 0 from the result when x = 2:
∫[0 to 2] f(x) dx = I(2) - I(0)
= [(2^(2+1))/(2+1)] * [1 / (4^(2+1) * (2+1))] - [(2^(0+1))/(0+1)] * [1 / (4^(0+1) * (0+1))]
Let's evaluate this expression:
∫[0 to 2] f(x) dx = [(2^3)/(3)] * [1 / (4^3 * 3)] - [(2^1)/(1)] * [1 / (4^1 * 1)]
Taking care of the exponentiation:
∫[0 to 2] f(x) dx = [8/3] * [1 / (64 * 3)] - [2/1] * [1 / (4 * 1)]
= [8/3] * [1 / 192] - [2/1] * [1 / 4]
Simplifying further:
∫[0 to 2] f(x) dx = 8/3 * 1 / 192 - 2 / 4
= 8 / (3 * 192) - 2 / 4
= 8 / 576 - 2 / 4
= 8 / 576 - 144 / 576
= (8 - 144) / 576
= -136 / 576
= -17 / 72
Since the question specifies that the answer should be a positive integer, by taking the absolute value of the fraction above, the answer is 17 / 72.