please answer and explain thank you 1. Answer the following statements with TRUE

Question

please answer and explain thank you

1. Answer the following statements with TRUE or FALSE. You don't have to justify your answers. Correct answer 3 points, wrong answer 3 points, no answer 0 points. (a) If a power series X anr" converges for all r 0, then it converges for all r E R. True False (b) If (T) is differentiable at all r E R, then the Taylor series of about o converges for all r E R. True False (c) If a power series SC an (T-1) converges for 4 then it also converges for r O True n False (d) If a power series SC an (T-1)" converges for r 4 then it also converges for T 4. True False True False o True False 0 (zero vector in R3), (g) If a and b are two vectors in R3 with a b 0 and a x b True False then a 0 or b (h) For three vectors a, b, c in R3, a x (b x c c x (a x b). True False

Explanation / Answer

a) Flase reason when x>1 then it will diverge because common ratio of geometric sequence is bigger than 1.

b)Yes that is true, because eqality between f(x) and taylor series only valid when series is converge.

c) Yes, this is also true, because when we plug x=4 then we get 3^n and when we plug x=-4 then we get (-1)^n*5^n , if a term containing 3^n is converge, then 5^n is also converge.

d) Yes this is also true, same reason (that is discussed in part c).

e) this is true, because multiplication of scalar in vector doesnot make any effect either it is multiply in any order, if we are taking dot product.

f) False, because it will change vector.

g) True.

h) true, because cross product of b and c is -a and cross prodcut of a and b is c so result is same for both side.

f

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