Reference no: EM132353954
Questions -
Q1. In each of the following, the general term of a sequence of real numbers is given. Determine if the sequence converges (that is, if the sequence has a finite limit) or if it diverges (that is, if the sequence doesn't have a finite limit) as n → ∞. If the sequence converges, determine its limit.
(a) an = 3n/(n+2).
(b) bn = (-1)n-1 n/(4n+2)
(c) cn = (-1)n-1 1/(2n+3)
(d) dn = (1+4)n/(1+4)n+1
Q2. Use the integral test to determine if each of the following series converges or diverges.
(a) n=2∑∞1/(n(ln n)5)
(b) n=2∑∞1/(n√(ln n))
Q3. Determine if the following series converge or diverge. Indicate clearly which test you are using to make a decision, and conduct the test.
(a) n=1∑∞1/(n+1)2
(b) n=1∑∞(-1)n-11/√(n+3)
(c) n=1∑∞1/(n2n)
Q4. (a) Express the indefinite integral ∫x5ln(1+x) dx as an infinite series.
(b) Use your result in (a) to approximate 0∫½x5ln(1+x)dx using the first four terms of the expansion.
Q5. For each of the following power series, determine the radius of convergence and the interval of convergence.
(a) n=0∑∞(-1)nx2n/2n+1
(b) n=0∑∞(-1)n xn/√(n2+1)
(c) n=0∑∞n!xn
Q6. (a) Sketch the graph of the curve whose parametric equations are
x = ¼t2 + t2, y = t2, for t ≥ 0.
(b) Find the point corresponding to t = 2, and mark it on the graph.
(c) Determine the slope of the line tangent to the graph of this curve when t = 2, and find the equation of this line.
Q7. (a) Sketch the graph of r = 4sin(2θ).
(b) Find the area inside the curve in the first quadrant, and use the symmetry of the graph to obtain the total area inside the curve.
Q8. For the curve whose parametric equations are: x = 1 + ln t, y = t2 + 2, t ≥ 1/e
(a) Sketch the graph of this curve.
(b) Find the point corresponding to t = 1 and mark it on the graph.
(c) Determine the equation of the line tangent to the curve at the point corresponding to t = 1. Sketch the graph of this line on the graph.
Q9. Sketch the graph of the curve whose polar equation is r = 10sin(5θ) and compute the area of the leaf that is completely contained in the first quadrant.
Q10. Find the limit of each of the following sequences if the sequence converges. If the sequence diverges, indicate so.
(a) an = 5n2/e2n
(b) bn = (1 - 2/n)n
(c) cn = 1 + 2/n
Q11. Test the series for convergence or divergence. If the series diverges, explain why; and if it converges, find the sum.
(a) n=0∑∞(-1)n(n+3)/(n+5)
(b) n=0∑∞(4n/52n)
Q12. (a) Use the Integral Test to determine whether n=3∑∞ 1/(n(ln n)) converges or diverges.
(b) Use the Integral Test to determine whether n=3∑∞1/(n(ln n)2) converges or diverges.
Q13. Use the Ratio Test to determine whether n=0∑∞(enn2)/n! converges or diverges.
Q14. Find the radius of convergence and the interval of convergence of each power series. Make sure that you test for convergence at the endpoints.
(a) n=1∑∞(-1)n+1(x-3)n/n4n
(b) n=0∑∞(-1)n(x+5)n/n+1
15. For the vectors in 3-space: u→ = i→ + 3j→ - 2k→, v→ = 2i→ - j→ + k→, determine:
(a) |u→| =
(b) |v→| =
(c) u→ x v→
(d) The angle between the vectors u→ = i→ + 3j→ - 2k→ and v→ = 2i→ - j→ + k→.
(e) A unit vector in the direction of u→ x v→.
Q16. Find the area of the parallelogram in 3-space spanned by the vectors u→ = (-1, 2, 5) and v→ (2, -3, 4).