EngMaths 245 - Week 1 - 11.1 to 11.2

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Last updated 9:31 AM on 7/31/26
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17 Terms

1
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Describe the condition for the convergence of a sequence

  • A sequence (ana_n)

  • For any ϵ\epsilon > 0:

    • if a natural number N exists with the following property:

      • n ≥\geq N

    • then | ana_n - L | < ϵ\epsilon

  • otherwise it diverges

<ul><li><p>A sequence ($$a_n$$)</p></li><li><p>For any $$\epsilon$$ &gt; 0:</p><ul><li><p>if a natural number N exists with the following property:</p><ul><li><p>n $$\geq$$ N</p></li></ul></li><li><p>then | $$a_n$$ - L | &lt; $$\epsilon$$ </p></li></ul></li><li><p>otherwise it diverges</p></li><li><p></p></li></ul><p></p>
2
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Describe the condition for the divergence of a sequence of a sequence

  • A sequence (ana_n)

  • For all numbers M:

    • if a natural number N exists with the following property:

      • n ≥\geq N

    • then ana_n ≥\geq M

<ul><li><p>A sequence ($$a_n$$)</p></li><li><p>For all numbers M:</p><ul><li><p>if a natural number N exists with the following property:</p><ul><li><p>n $$\geq$$ N</p></li></ul></li><li><p>then $$a_n$$ $$\geq$$ M</p></li></ul></li></ul><p></p>
3
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Derive the proof relating the limit of a function to the limit of a sequence

  • if lim⁡x→∞f(x)=L\lim_{x \to \infty} f(x) = L and

    • an=f(n)a_n = f(n) then

    • lim⁡n→∞an=L\lim_{n \to \infty} a_n = L

<ul><li><p>if $$\lim_{x \to \infty} f(x) = L$$ and</p><ul><li><p>$$a_n = f(n)$$ then</p></li><li><p>$$\lim_{n \to \infty} a_n = L$$ </p></li></ul></li></ul><p></p>
4
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State the limit laws referring to the limit of two sequences that have the following operations applied to each other:

  • summed

  • subtracted

  • divided

  • multiplied

% 1) Sum Law

lim⁡(an+bn)=lim⁡(an)+lim⁡(bn)\lim (a_n + b_n) = \lim (a_n) + \lim (b_n)

% 2) Difference Law

lim⁡(an−bn)=lim⁡(an)−lim⁡(bn)\lim (a_n - b_n) = \lim (a_n) - \lim (b_n)

% 4) Product Law

lim⁡(anbn)=lim⁡(an)lim⁡(bn)\lim (a_n b_n) = \lim (a_n) \lim (b_n)

% 5) Division (Quotient) Law

lim⁡(an/bn)=lim⁡(an)lim⁡(bn),lim⁡(bn)≠0\lim (a_n / b_n) = \frac{\lim(a_n)}{\lim(b_n)}, \quad \lim(b_n) \neq 0

5
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State the limit law regarding the limit of the product of a sequence and a constant

  • lim⁡(c⋅an)=c⋅lim⁡(an)\lim (c\cdot a_n) = c\cdot \lim(a_n)

6
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State the limit law regarding the limit of the power of a sequence and a variable p > 0, a_n > 0

  • lim⁡(anp)=(lim⁡(an))p\lim (a_n^p) = (\lim(a_n))^p

7
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Describe the squeeze theorem regarding sequences

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8
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State the limit law regarding lim⁡n→∞∣an∣=0\lim_{n \to \infty} |a_n| = 0

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9
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State the limit law of sequences relating the limit of a sequence and a function of that sequence as an input

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10
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Define a monotone sequence

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11
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Define a bounded sequence

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12
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Relate the monotone and bounded properties of a sequence to it convergence

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13
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Describe and define what is meant by the series and partial sums of a sequence

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14
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Relate the convergence of a sequence to its series

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15
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Derive the equation for the nth sum of a geometric series

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16
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Derive the condition for the convergence of a geometric series

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17
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Derive and state the series laws related to:

  • addition

  • subtraction

  • constant

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