Worksheet 2 (07/24/2015)
Real Analysis II
•Multivariable calculus and series: Derivative. Gradient, curl and divergence.
Minimum/Maximum. Lagrange multiplier. Jacobian determinant. Line inte-
gral. Surface integral. Absolute and conditional convergence. Rearrangement.
Summation by parts. Power series. Multiplication of series. Sequence/series of
functions.
•Relating integrals over interior and boundary, and tests for conver-
gence: Inverse and implicit function theorem. Rank theorem. Green, Stokes,
Gauss theorem. Contraction mapping theorem. Limit comparison. Integral com-
parison. Ratio and root tests. Alternating series. Dirichlet test.
•Common Approaches:
–Root test is better than ratio test.
–Use implicit function theorem to extend solution of differential equation.
–Use interpolation to turn a sequence into a function. Use power series to
turn a series into a function.
1 Discuss the solvability of the differential equation
(exsin y)(y0)3+ (excos y)y0+eytan x= 0
with the initial condition y(0) = 0. Does a solution exist in some interval about 0? If
so, is it unique?
2 Let M2×2be the space of 2 ×2 matrices over Ridentified with R4. Define a function
F:M2×23X→X+X2∈M2×2. Prove that the range of Fcontains a neighborhood
of the origin.
3 Let aand x0be positive numbers. Recursively define a sequence {xn}n>0by
xn:= 1
2xn−1+a
xn−1. Prove that this sequence converges. Find the limit.
4 Let {bn}n>0be a sequence of real numbers with bk≥bk+1 for all kwith lim
k→∞
bk= 0.
Prove that the power series P∞
k=1 bkzkconverges for |z| ≤ 1 with z6= 1.
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