Math 143: Calculus III Midterm I Practice Problems
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Math 143: Calculus III
Midterm I Practice Problems
October 4th, 2022
1. (10 points) Determine whether the following sequences converge or diverge. If they converge, find their limit. If they diverge, state whether they diverge to +● , -● or because they oscillate. Justify and show all your work.
(a)
an = cos(n)
(b)
an = sin ╱ 、
(c)
an = ln (2n + 3) - ln (3n + 2)
(d)
an = cos ╱ 、
(e)
cos(n)
n
(f)
an = (-1)n
(g)
an = ╱ 、 n
(h)
(-1)n
an = 6n
(i)
an = ln ╱ 、
(j)
ln(n)
n
(k)
an =
(l)
an =
en - 1
(m)
sin(n)
en
(n)
sin(n)
n!
(o)
an = ln (3n2 + 1) - ln (n + 1)
2. (10 points) Determine whether the following series converge or diverge. If a series converges, find its sum. Justify and show all your work. Name any test you are using.
(a)
良 ╱ 、 n
(b)
良 (-3)n + 3n
10n
n≠〇
(c)
良 (-1)n
6n
n≠〇
(d)
良 ╱sin ╱ 、 - sin ╱ 、、
(e)
1
n(n + 3)
n≠1
(f)
良 3n
(g)
n |
ln(n) |
3. (10 points) Determine whether the following series converge or diverge. Justify and show all your work. Name any test you are using.
(a)
良 n
n3 + 5n
n≠1
(b)
良 arctan(n)
n1.2 - 6
n≠1
(c)
良 6n + n
5n - 9
n≠1
(d)
n
n!
n≠1
(e)
良 (-1)n
n≠1 n + 8
(f)
1
n(ln n)
n≠2
(g)
良 ln(n)
2
(h)
良 n
2n + 5
n≠1
(i)
良 (-1)n
n≠2 ln n
(j)
良 ╱ 、2
(k)
良 ln ╱ 、
(l)
4
n1.1
n≠1
(m)
良 (-1)nn
ln n
(n)
1
en
n≠1
(o)
1
n(n + 1)
n≠1
(p)
1
n≠1 n + 1
(q)
良
e ìn - e ì _n+1)
n≠1
(r)
n
n3 + 5n
n≠1
(s)
1
n≠1 n!
4. (10 points) Use the integral test to determine whether the following series converges or diverges. To get full credit you must use the integral test.
(a)
(b)
1
n
n≠1
(c)
1
n(ln n)2
n≠2
(d)
2n
n2 + 5
n≠1
2022-09-28