Problem 44513. Add all the numbers between two limits (inclusive)

Created by David Verrelli

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{"relationships":[{"relationshipType":"http://schemas.mathworks.com/matlab/code/2013/relationships/document","targetMode":"","relationshipId":"rId1","target":"/matlab/document.xml"},{"relationshipType":"http://schemas.mathworks.com/matlab/code/2013/relationships/output","targetMode":"","relationshipId":"rId2","target":"/matlab/output.xml"}],"parts":[{"partUri":"/matlab/document.xml","relationship":[],"contentType":"application/vnd.mathworks.matlab.code.document+xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\"?>\n<w:document xmlns:w=\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\"><w:body><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>In this problem you must add up \"all of the numbers\" between two specified limits,</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>a</w:t></w:r><w:r><w:t> and</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>b</w:t></w:r><w:r><w:t>, in which</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>a</w:t></w:r><w:r><w:t> ≤</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>b</w:t></w:r><w:r><w:t>. However, the practical interpretation of \"all of the numbers\" will depend upon the specified</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"https://au.mathworks.com/help/matlab/numeric-types.html\"><w:r><w:t>data type</w:t></w:r></w:hyperlink><w:r><w:t>,</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>dt</w:t></w:r><w:r><w:t>.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>Mathematically speaking, if</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>a</w:t></w:r><w:r><w:t> &lt;</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>b</w:t></w:r><w:r><w:t> then the required sum constitutes an</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>infinite series</w:t></w:r><w:r><w:t> that does not converge (i.e. the required sum would be infinity). For example, if</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>a=1</w:t></w:r><w:r><w:t> and</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>b=2</w:t></w:r><w:r><w:t> then we could capture</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>some</w:t></w:r><w:r><w:t> of those numbers through the series lim n→∞ ⁿ∑ᵢ₌₁{1 + (1/i)} = lim n→∞ {n + ⁿ∑ᵢ₌₁(1/i)} ≈ lim n→∞ {n + γ + ln(n)}, using properties of the harmonic series in the last approximation.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>But MATLAB cannot represent numbers with</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>infinite</w:t></w:r><w:r><w:t> precision. In fact, the precision is determined by the specified</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"https://au.mathworks.com/help/matlab/numeric-types.html\"><w:r><w:t>data type</w:t></w:r></w:hyperlink><w:r><w:t>. For instance, if</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>dt = 'single'</w:t></w:r><w:r><w:t>, then with</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>a=1</w:t></w:r><w:r><w:t> and</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>b=2</w:t></w:r><w:r><w:t> the summation would comprise the series {(1) + (1+1×2⁻²³) + (1+2×2⁻²³) + (1+3×2⁻²³) + ... + (2−2×2⁻²³) + (2−1×2⁻²³) + (2)} = 12582913.5, which is finite.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>Another example:</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"code\"/></w:pPr><w:r><w:t><![CDATA[ % INPUT\n a = 10\n b = 12\n dt = 'int16'\n % OUTPUT\n s = 33 % = 10 + 11 +12]]></w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>So please add up all the numbers between two limits (inclusive), subject to the precision indicated by the specified</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"https://au.mathworks.com/help/matlab/numeric-types.html\"><w:r><w:t>data type</w:t></w:r></w:hyperlink><w:r><w:t>.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:rPr><w:i/></w:rPr><w:t>NOTE</w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t> </w:t></w:r><w:r><w:rPr><w:b/><w:i/></w:rPr><w:t>1</w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>: Terminal values</w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t> </w:t></w:r><w:r><w:rPr><w:i/><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>a</w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t> and</w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t> </w:t></w:r><w:r><w:rPr><w:i/><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>b</w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t> are whole numbers in every case (albeit implicitly defined as of the</w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t> </w:t></w:r><w:r><w:rPr><w:i/><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>double</w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t> data type); they can be positive or negative. However, values -1&lt;x&lt;+1 are never included in the summations.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:rPr><w:i/></w:rPr><w:t>NOTE</w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t> </w:t></w:r><w:r><w:rPr><w:b/><w:i/></w:rPr><w:t>2</w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>: All data types specified in the input</w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t> </w:t></w:r><w:r><w:rPr><w:i/><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>dt</w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t> shall be</w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"https://au.mathworks.com/help/matlab/numeric-types.html\"><w:r><w:rPr><w:i/></w:rPr><w:t>numeric</w:t></w:r></w:hyperlink><w:r><w:rPr><w:i/></w:rPr><w:t>.</w:t></w:r></w:p></w:body></w:document>"},{"partUri":"/matlab/output.xml","contentType":"text/xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\" ?><embeddedOutputs><metaData><evaluationState>manual</evaluationState><layoutState>code</layoutState><outputStatus>ready</outputStatus></metaData><outputArray type=\"array\"/><regionArray type=\"array\"/></embeddedOutputs>"}]}

In this problem you must add up "all of the numbers" between two specified limits, <tt>a</tt> and <tt>b</tt>, in which <tt>a</tt> ≤ <tt>b</tt>. However, the practical interpretation of "all of the numbers" will depend upon the specified <a href = "https://au.mathworks.com/help/matlab/numeric-types.html">data type</a>, <tt>dt</tt>.Mathematically speaking, if <tt>a</tt> &lt; <tt>b</tt> then the required sum constitutes an infinite series that does not converge (i.e. the required sum would be infinity). For example, if <tt>a=1</tt> and <tt>b=2</tt> then we could capture some of those numbers through the series 
lim n→∞ ⁿ∑ᵢ₌₁{1 + (1/i)} = lim n→∞ {n + ⁿ∑ᵢ₌₁(1/i)} ≈ lim n→∞ {n + γ + ln(n)}, using properties of the harmonic series in the last approximation.But MATLAB cannot represent numbers with infinite precision. In fact, the precision is determined by the specified <a href = "https://au.mathworks.com/help/matlab/numeric-types.html">data type</a>. For instance, if <tt>dt = 'single'</tt>, then with <tt>a=1</tt> and <tt>b=2</tt> the summation would comprise the series {(1) + (1+1×2⁻²³) + (1+2×2⁻²³) + (1+3×2⁻²³) + ... + (2−2×2⁻²³) + (2−1×2⁻²³) + (2)} = 12582913.5, which is finite.Another example:<pre> % INPUT
 a = 10
 b = 12
 dt = 'int16'
 % OUTPUT
 s = 33 % = 10 + 11 +12</pre>So please add up all the numbers between two limits (inclusive), subject to the precision indicated by the specified <a href = "https://au.mathworks.com/help/matlab/numeric-types.html">data type</a>.NOTE 1: Terminal values <tt>a</tt> and <tt>b</tt> are whole numbers in every case (albeit implicitly defined as of the <tt>double</tt> data type); they can be positive or negative. However, values -1&lt;x&lt;+1 are never included in the summations.NOTE 2: All data types specified in the input <tt>dt</tt> shall be <a href = "https://au.mathworks.com/help/matlab/numeric-types.html">numeric</a>.

In this problem you must add up "all of the numbers" between two specified limits, |a| and |b|, in which |a| ≤ |b|. However, the practical interpretation of "all of the numbers" will depend upon the specified <https://au.mathworks.com/help/matlab/numeric-types.html data type>, |dt|.

Mathematically speaking, if |a| < |b| then the required sum constitutes an _infinite series_ that does not converge (i.e. the required sum would be infinity). For example, if |a=1| and |b=2| then we could capture _some_ of those numbers through the series 
lim n→∞ ⁿ∑ᵢ₌₁{1 + (1/i)} = lim n→∞ {n + ⁿ∑ᵢ₌₁(1/i)} ≈ lim n→∞ {n + γ + ln(n)}, using properties of the harmonic series in the last approximation.

But MATLAB cannot represent numbers with _infinite_ precision. In fact, the precision is determined by the specified <https://au.mathworks.com/help/matlab/numeric-types.html data type>. For instance, if |dt = 'single'|, then with |a=1| and |b=2| the summation would comprise the series {(1) + (1+1×2⁻²³) + (1+2×2⁻²³) + (1+3×2⁻²³) + ... + (2−2×2⁻²³) + (2−1×2⁻²³) + (2)} = 12582913.5, which is finite.

Another example:

% INPUT
 a = 10
 b = 12
 dt = 'int16'
 % OUTPUT
 s = 33         %  = 10 + 11 +12

So please add up all the numbers between two limits (inclusive), subject to the precision indicated by the specified <https://au.mathworks.com/help/matlab/numeric-types.html data type>.

_NOTE *1*: Terminal values |a| and |b| are whole numbers in every case (albeit implicitly defined as of the |double| data type); they can be positive or negative. However, values -1<x<+1 are never included in the summations._

_NOTE *2*: All data types specified in the input |dt| shall be <https://au.mathworks.com/help/matlab/numeric-types.html numeric>._

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Tim on 6 Feb 2018

For single (and double) the test suite result for the sum from 2 to 3 is smaller than that for 1 to 2--can that be right?

Tim on 6 Feb 2018

I think I commented too soon again--never mind.

David Verrelli on 7 Feb 2018

No worries, Tim. I found it to be an interesting result too :-) . . . —DIV

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Problem 44513. Add all the numbers between two limits (inclusive)

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