ASSIGNMENT X

(series and sequence, circle and radius of convergence, Taylor series, classification of singularities)

1. The nature of singularity of the function (𝑧 − sin 𝑧)/𝑧2 is

(a) isolated singularity (b) removable singularity (c) pole (d) essential singularity

2. The nature of singularity of the function 1/(cos 𝑧 − sin 𝑧) is

(a) isolated singularity (b) removable singularity (c) pole (d) essential singularity

3. The nature of singularity of the function 𝑒1

𝑧/𝑧2 is

(a) isolated singularity (b) removable singularity (c) pole (d) essential singularity

4. The radius of convergence of the series ∑(2𝑛)!

(𝑛!)2∞𝑛=0 (𝑧 − 3𝑖)𝑛 is

(a) 2 (b) 1/2 (c) 4 (d) 1/4

5. The radius of convergence of the series ∑ (1 + (−1)𝑛 +1

2𝑛) 𝑧𝑛∞

𝑛=0 is

(a) 1/2 (b) 1 (c) 2 (d) 1/3

6. The Taylor series expansion of 𝑓(𝑧) =1

1−𝑧 about 𝑧0 = 0 is

(a) 𝟏 + 𝒛 + 𝒛𝟐 + 𝒛𝟑 +⋯ (b) 1 − 𝑧 + 𝑧2 − 𝑧3 +⋯

(c) 1 + 𝑧 +𝑧2

2!+

𝑧3

3!+⋯ (d) 1 − 𝑧 +

𝑧2

2!−

𝑧3

3!+⋯

7. The radius of convergence of the Taylor series in Q.6 is

(a) 0 (b) 1 (c) 2 (d) 1/2

8. The Taylor series expansion of 𝑓(𝑧) = sin 𝑧 about 𝑧0 = 𝜋/4 is

(a) 1

√2[1 + (𝑧 −

𝜋

4) +

(𝑧−𝜋

4)2

2!+

(𝑧−𝜋

4)3

3!+⋯] (b)

𝟏

√𝟐[𝟏 + (𝒛 −

𝝅

𝟒) −

(𝒛−𝝅

𝟒)𝟐

𝟐!−

(𝒛−𝝅

𝟒)𝟑

𝟑!+⋯]

(c) 1

2[1 + (𝑧 −

𝜋

4) −

(𝑧−𝜋

4)2

2!−

(𝑧−𝜋

4)3

3!+⋯] (d)

1

√2[1 + (𝑧 −

𝜋

4) −

(𝑧−𝜋

4)2

2!+

(𝑧−𝜋

4)3

3!+⋯]

9. The region of convergence of the Taylor series in Q.8 is

(a) |𝑧| < 1 (b) |𝑧| ≤ 𝜋/4 (c) |𝑧| < 2 (d) |𝒛| < ∞

10. The Taylor series expansion of 𝑓(𝑧) = ln(1 + 𝑧) about the origin is

(a) 𝑧 + 𝑧2 − 𝑧3 + 𝑧4 +⋯ (b) 𝒛 −𝒛𝟐

𝟐+

𝒛𝟑

𝟑−

𝒛𝟒

𝟒+⋯

(c) 𝑧 +𝑧2

2!+

𝑧3

3!+

𝑧4

4!… (d) 𝑧 −

𝑧2

2!+

𝑧3

3!−

𝑧4

4!+⋯

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