Mathematics - Mathematical Analysis Subsequences and Convergence
mathematics·@drifter1·
0.000 HBDMathematics - Mathematical Analysis Subsequences and Convergence
<html> <p><img src="https://upload.wikimedia.org/wikipedia/commons/thumb/6/62/Cauchy_sequence_illustration.svg/350px-Cauchy_sequence_illustration.svg.png" width="350" height="195"/></p> <p> Hello its me again drifter1. Today we continue with <strong>Mathematical Analysis</strong> getting into <strong>Subsequences </strong>and the <strong>Convergence </strong>of Sequences. I suggest you to check out my previous post about the Basics of Sequences <a href="https://steemit.com/mathematics/@drifter1/mathematics-mathematical-analysis-sequence-basics">here </a>before getting into this post! So, without further do, let's get straight into it!</p> <h2>Subsequence:</h2> <p> Subsequence of an sequence (an) is every sequence (bn) with generic term b(n) = a(kn), for every natural number n and with k(n) being a strictly increasing sequence of natural numbers. That's why we write a subsequence if (an) as (akn) for the specific sequence (kn).</p> <p><strong>Example:</strong></p> <p><strong> </strong>If we take only the odd terms a(1), a(3), ..., a(2n-1), ... of a sequence (an), then end up with a subsequence (bn) with generic term b(n) = a(2n-1), for every natural number n and for a sequence (kn) with generic term k(n) = 2n-1. We could do the same with even terms and end up with c(n) = a(2n) and so l(n) = 2n. The "sum" of those two subsequences equals the given sequence. So, this means that we can split a sequence into many subsequences and the "sum" of those will give us our first sequence.</p> <h3>Subsequence Boundary and Monotony:</h3> <ul> <li>If a sequence (an) is bounded (upper and lower) then every subsequence (akn) of (an) is also bounded.</li> <li>If a sequence (an) is upper (or lower) bounded then every subsequence (akn) of (an) is also upper (or lower) bounded.</li> <li>If at least one of the subsequences of (an) is not bounded then (an) is also not bounded.</li> <li>If at least one of the subsequence of (an) is not upper (or lower) bounded then (an) is also not upper (or lower) bounded.</li> <li>If a sequence (an) is strictly monotonic (increasing or decreasing) then every subsequence (akn) is also strictly monotonic (increasing or decreasing) .</li> </ul> <h2>Sequence Convergence:</h2> <p>A sequence (an) is called a <strong>null sequence</strong> when lim n -> +∞ (an) = 0.</p> <p>For <strong>example </strong>the sequence a(n) = 1/n is strictly increasing, but also a null sequence.</p> <p>So, the convergence has to do with the limit to infinity.</p> <p> If the <strong>limit is equal to a real number l </strong>then we say that the<strong> sequence converges</strong> (is convergent) and this number l is called the<strong> limit of the sequence</strong>. So, <em>lim n -> +∞ (an) = l</em> . When the <strong>limit doesn't exist or equals +-∞</strong> then the<strong> sequence deverges</strong> (is devergent).</p> <p><br></p> <p><strong>Examples:</strong></p> <p><strong>1. </strong>a(n) = 2 + 1/n</p> <p>lim n -> +∞ a(n) = lim n -> +∞ [2 + 1/n] = 2.</p> <p>So, the sequence converges to 2.</p> <p><br></p> <p><strong>2. </strong>b(n) = (4n^2 -3n + 4) / (n^2 +1)</p> <p>lim n -> +∞ b(n) = lim n -> +∞ [ (4n^2 -3n + 4) / (n^2 +1)] = 4/1 = 4.</p> <p>Cause this is a polynomial limit with same degree on numerator and denominator.</p> <p>So, the sequence converges to 4.</p> <p><br></p> <p><strong>Properties:</strong></p> <ul> <li>The limit l of a sequence (an) is unique if it exists (sequence is convergent).</li> <li>If a sequence (an) is convergent then it also is bounded. The opposite is not true!</li> <li>If a sequence (an) is not bounded then it also doesn't converge.</li> <li>If a sequence (an) converges to l then every subsequence (akn) of (an) also converges to l.</li> <li>If 2 or more subsequences of (an) converge to a different number l then the sequence (an) doesn't converge.</li> <li>If a sequence (an) gets cut down to many subsequences that converge to the same limit l, then l is also the limit of (an).</li> </ul> <p>If (an) and (bn) are 2 convergent sequences then:</p> <ol> <li>lim n -> +∞ [a(n) +- b(n)] = a + b, where a and b are the limits of (an) and (bn).</li> <li>lim n -> +∞ [a(n) */ b(n)] = a */ b, where a and b are the limits of (an) and (bn).</li> <li>lim n -> +∞ [1/a(n)] = 1/a, with a!=0 and a(n)!=0 for every natural n. </li> <li>lim n -> +∞ [c*a(n)] = c * a, with c being a real number and a the limit of (an)</li> </ol> <p>The same properties can be applied to more than 2 sequences that converge.</p> <h3>Some more things that we can proof:</h3> <p><br></p> <p><strong>If (an) is a null sequence and (bn) is bounded</strong> then:</p> <p><em>lim n -> +∞ [a(n) * b(n)] = 0 </em></p> <p><br></p> <p><strong>If (an) is monotonic and bounded </strong>then:</p> <ul> <li>If (an) is strictly increasing then it also is upper bounded and converges to the supremum bound</li> <li>If (an) is strictly decreasing then it also is lower bounded and converges to the infimum bound</li> </ul> <p><br></p> <p><strong>If(an) is a convergent sequence with limit a</strong> then:</p> <ul> <li>the <strong>absolute sequence </strong>|(an)| converges to |a|. The opposite is not true!</li> <li>The limit of the <strong>k-root</strong> of |(an)| equals the k-root of |a|, for a natural number k.</li> </ul> <p><br></p> <p><strong>Squeeze Theorem or Sandwich Theorem/Rule for sequences:</strong></p> <p>Suppose the sequences (an), (bn) and (cn) with bn <= an <= cn for every natural number n.</p> <p> If (bn) and (cn) converge to the same limit l in R and so lim n -> +∞ b(n) = l = lim n -> +∞ c(n) then the limit of the sequence (an) is also equal to l. So, lim n -> +∞ a(n) = l.</p> <p><br></p> <p><strong>Convergence Criterion:</strong></p> <p> If (an) and (bn) are sequences with |a(n)| <= |b(n)| for every natural n, then if (bn) a null sequence then (an) is also a null sequence and so converges.</p> <p><br></p> <p><br></p> <p>And this is actually it for today and I hope you enjoyed it!</p> <p>Next time we will talk about some special and devergent sequences.</p> <p>Until next time...Bye!</p> </html>
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