Type your question here Consider the series (a) The graph below shows the relati

Question

Type your question here

Consider the series (a) The graph below shows the relationship between the function f(x) = e1/x/x2 and the terms of an infinite series . The function f(x) is continuous, positive and decreasing over the interval[l, infinity ). Evaluate and show the series converges. Use your answer to part (a) and the given graph to give an upper bound for the sum of the series. Use the remainder estimate for the integral test to determine the bounds for the remainder, R3, of the third partial sum

Explanation / Answer

(a)

We have to compare it with the integral between 1 and +infinity :

Since e^(1/x)/x^2 = -e^(1/x) * (1/x)' then :

int(1<x<+infinity) e^(1/x)/x^2 dx = [-e^(1/x)](1<x<+inf) = e-1

So the series converges by the integral test

(b)

For the graph you see that the sum of area of rectangle a2+...+an <= is less than area under f(x) ( in [1,+infinity) )

So a1+..+an <= a1 + (e-1) and a1 = e so a1+...+an <= 2e-1

So an upper bound is 2e-1

(c) We have int(4<x<+infinity) f(x)dx < R_3 < int(3<x<+infinity) f(x)

[-e^(1/x)](4<x<+inf] = e^(1/4)-1 = 0.2840

[-e^(1/x)](3<x<+inf] = e^(1/3)-1 = 0.3956

So R_3 is in [0.2840,0.3956]

Related Questions

Navigate

• Browse (All)
• Browse T
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.