Determine whether the following statements are true or false. If it is true, exp

Question

Determine whether the following statements are true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. By CT, integral^0_-1 e^1/3/x^3 dx is a convergent integral. If integral^infinity_7 f(x)dx converges and let c R, then integral^infinity_7 cf(x)dx also converges. Let a R. If integral^infinity_a f(x)dx and integral^infinity_a g(x)dx are both convergent, then integral^infinity_a [f(x) + g(x)]dx is convergent. Let a R. If integral^infinity_a f(x)dx and integral^infinity_a g(x)dx are both divergent, then integral^infinity_a [f(x) + g(x)]dx is divergent. Let a R. If 0 le f(x) le g(x) for all x [a, infinity) and integral^infinity_a g(x) dx diverges, then integral^infinity_a f(x)dx also diverges.

Explanation / Answer

(a) TRUE : Because e^(1/x) is decreasing on (-1,0) and x^3 increasing on (-1,0)

So, decreasing / increasing always converges.

(b) TRUE : BEcause If we multiply any converges series with real number. It's converges or no change in nature of series.

(c) TRUE

Becuase sum of two convergent series always converges.

(d) TRUE

Because sum of two divergent series always divergent.

(e) FALSE

It's not neccessary if function is small it should diverges also.

Related Questions

Navigate

• Browse (All)
• Browse D
• Subjects

---

---

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