Questions from Calculus

Q: Verify the formula ∑ x=1 n x = n

Verify the formula ∑ x=1 n x = n(n + 1) / 2 for n = 10, 50, and 100.

Q: The sum of the first n odd numbers is n2; that

The sum of the first n odd numbers is n2; that is, ∑x=1 n (2x - 1) = n2. Verify this formula for n = 5, 10, and 25.

Q: Convince yourself that the equation is correct by summing up the first

Convince yourself that the equation is correct by summing up the first 999 terms of the infinite series and comparing the sum with the value on the right. ∑x=1 ∞ 1/x2 = π2/6

Q: Convince yourself that the equation is correct by summing up the first

Convince yourself that the equation is correct by summing up the first 999 terms of the infinite series and comparing the sum with the value on the right. ∑x=1 ∞ (-1)x+1 / x = ln 2

Q: Determine the sums of the following geometric series when they are convergent

Determine the sums of the following geometric series when they are convergent. 1 + 1/6 + 1/62 + 1/63 + 1/64 …

Q: Use a second Taylor polynomial at x = 0 to estimate the

Use a second Taylor polynomial at x = 0 to estimate the area under the curve y = √(cos x) from x = -1 to x = 1. (The exact answer to three decimal places is 1.828.)

Q: Use the integral test to determine whether the infinite series is convergent

Use the integral test to determine whether the infinite series is convergent or divergent. (You may assume that the hypotheses of the integral test are satisfied.) ∑k=2 ∞ 1/(k - 1)3

Q: Use the integral test to determine whether the infinite series is convergent

Use the integral test to determine whether the infinite series is convergent or divergent. (You may assume that the hypotheses of the integral test are satisfied.) ∑k=0∞ 7/(k + 100)

Q: Use the integral test to determine whether the infinite series is convergent

Use the integral test to determine whether the infinite series is convergent or divergent. (You may assume that the hypotheses of the integral test are satisfied.) ∑k=1∞ 2/(5k – 1)

Q: Use the integral test to determine whether the infinite series is convergent

Use the integral test to determine whether the infinite series is convergent or divergent. (You may assume that the hypotheses of the integral test are satisfied.) ∑k=2∞ 1/k√(ln k)