Homework 6Solutions

Exercise 1. The air in a room with volume 180 m3 contains 0.15% carbon dioxideinitially. Fresher air with only 0.05% carbon dioxide flows into the room at a rate of2 m3/min and the mixed air flows out at the same rate. Find the percentage of carbondioxide in the room as a function of time. What happens in the long run?

1

Exercise 2. Solve the differential equation.

1. (x2 + 1)y′ = xy

2.dz

dt+ et+z = 0

Exercise 3. Find the solution of the differential equation that satisfies the given initialcondition.

1.dy

dx=

y cosx

1 + y2, y(0) = 1

2. x cosx = (2y + e3y)y′, y(0) = 0

2

Exercise 4. Determine whether the sequence converges or diverges. If it converges, findthe limit.

1. an =3 + 5n2

n+ n2

2. an =

√n

1 +√n

3. an =n

1 +√n

4. an =(−1)n−1nn2 + 1

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Exercise 5. Determine whether the series is convergent or divergent. If it is convergent,find its sum.

1.

∞∑n=1

1

2n

2.

∞∑k=1

k2

k2 − 1

3.∞∑n=1

1 + 2n

3n

4

Exercise 6. Determine whether the series is convergent or divergent, using either theComparison or Integral Test as appropriate.

1.

∞∑n=2

1

n lnn

2.∞∑n=1

cos2 n

n2 + 1

3.

∞∑n=1

4 + 3n

2n

4.

∞∑n=1

ne−n

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