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Fundamentals of Mathematical Analysis
 
Songs in Easy English
 
› 1 Preliminaries
 
› 2 Real Numbers
Between any two rationals there is an irrational.
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› 3 Sequences
30 approximating - √2, φ ┆ 311 Limit (Sequence) ┆ 312 Rules (Sequence) ┆ 313 Sandwich rule (Sequence) ┆ 314 Composite rule (Sequence) ┆ 32 Expansion of Operators and Numbers ┆ 321 Power rule (Sequence) ┆ 322 Basic null sequences ┆ 322_ Binomial Expansion ┆ 335 Subsequence ┆ 341 Principle of monotone sequences ┆ 342 Bonzano-Weierstrass theorem ┆ 342 Bonzano-Weierstrass theorem ┆ 343 Cauchy sequence ┆ 34_ ｢(1+1/n)˄n｣ is convergent ┆ harmonic numbers
 
› 4 Series
4 Series ┆ 411 Geometric Series ┆ 411 Geometric Series ┆ 415 p-series ┆ 415 Harmonic series ┆ 415 ｢1/n² Σn=1,∞｣ ┆ 421 First comparison test ┆ 422 Second comparison test ┆ A first look at series ┆ Power Series ┆ Series
 
› 5 Continuous Functions
511 Limit ┆ 512 Limit ┆ 513 Rules ┆ 522 Continuous ┆ Practice
 
› 6 Differentiation
611 Differentiable ┆ 612 differentiable ⇒ continuous
 
› 7 Integration
f is continuous ⇒ ❶ f is integrable ❷ f(x)ᐃx is differentiable ❸ (f(x)ᐃx)ᐁx = f
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• ｢↑x/x Ƚx￫0｣ = 1
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