{"id":653,"date":"2020-06-18T01:53:40","date_gmt":"2020-06-18T01:53:40","guid":{"rendered":"http:\/\/mathfun4kids.com\/mlog\/?p=653"},"modified":"2024-10-25T02:44:07","modified_gmt":"2024-10-25T06:44:07","slug":"mbmt-2020-problem-44","status":"publish","type":"post","link":"https:\/\/mathfun4kids.com\/mlog\/?p=653","title":{"rendered":"MBMT 2020 &#8211; Problem 44"},"content":{"rendered":"\n<p>Let $a_n=\\sum_{d|n}\\dfrac{1}{2^{d+\\frac{n}{d}}}$. In other words, $a_n$ is the sum of $\\dfrac{1}{2^{d+\\frac{n}{d}}}$ over all divisers $d$ of $n$. Find $$\\dfrac{\\sum_{k=1}^{\\infty}ka_k}{\\sum_{k=1}^{\\infty}a_k}=\\dfrac{a_1+2a_2+3a_3+&#8230;}{a_1+a_2+a_3+&#8230;}$$<\/p>\n\n\n\nClick <a onclick=\"toggle_visibility('mbmt-2020-44');\">here<\/a> for the solution.\n<div id=\"mbmt-2020-44\" style=\"display:none\">\n\n\n\n<p><strong>Solution:<\/strong> For the denominator, we have:\n$$\\begin{align}\n\\sum_{n=1}^{\\infty}a_n &amp; = \\sum_{n=1}^{\\infty}\\sum_{d|n}\\dfrac{1}{2^{d+\\frac{n}{d}}} = \\sum_{d=1}^{\\infty}\\sum_{n\\ge1,d|n}\\dfrac{1}{2^{d+\\frac{n}{d}}} \\\\\n&amp; = \\sum_{d=1}^{\\infty}\\dfrac{1}{2^d}\\sum_{k=1}^{\\infty}\\dfrac{1}{2^k} \\\\\n&amp;=\\sum_{d=1}^{\\infty}\\dfrac{1}{2^d}\\cdot 1=1 \\\\\n\n\\end{align}\n$$\n<\/p>\n\n\n\n<p>For the numerator, we have:\n$$\\begin{align}\n\\sum_{n=1}^{\\infty}na_n &amp; = \\sum_{n=1}^{\\infty}n\\sum_{d|n}\\dfrac{1}{2^{d+\\frac{n}{d}}} = \\sum_{d=1}^{\\infty}\\sum_{n\\ge1,d|n}\\dfrac{n}{2^{d+\\frac{n}{d}}} \\\\\n&amp; = \\sum_{d=1}^{\\infty}\\sum_{n\\ge1,d|n}\\dfrac{d\\cdot\\dfrac{n}{d}}{2^{d+\\frac{n}{d}}} = \\sum_{d=1}^{\\infty}\\dfrac{d}{2^d}\\sum_{k=1}^{\\infty}\\dfrac{k}{2^k} \\\\\n&amp; =\\sum_{d=1}^{\\infty}\\dfrac{d}{2^d}\\cdot 2=2\\cdot 2=4 \\\\\n\n\\end{align}\n$$\n<\/p>\n\n\n\nTherefore, the answer to the question is: $\\boxed{4}$\n\n\n\n<p><strong>Note<\/strong> For the sum of the <a rel=\"noreferrer noopener\" href=\"https:\/\/en.wikipedia.org\/wiki\/Arithmetico%E2%80%93geometric_sequence\" target=\"_blank\">arithmetico\u2013geometric<\/a>\u00a0series: $S=\\sum_{n=1}^{\\infty}\\dfrac{n}{2^n}$, we have $$\\begin{align} S&amp;=2\\cdot S &#8211; S =2\\sum_{n=1}^{\\infty}\\dfrac{n}{2^n}-\\sum_{n=1}^{\\infty}\\dfrac{n}{2^n} =\\sum_{n=1}^{\\infty}\\dfrac{n}{2^{n-1}}-\\sum_{n=1}^{\\infty}\\dfrac{n}{2^n} =\\sum_{n=0}^{\\infty}\\dfrac{1}{2^n}=2 \\\\ \\end{align} $$<\/p>\n\n\n\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Let $a_n=\\sum_{d|n}\\dfrac{1}{2^{d+\\frac{n}{d}}}$. In other words, $a_n$ is the sum of $\\dfrac{1}{2^{d+\\frac{n}{d}}}$ over all divisers $d$ of $n$. Find $$\\dfrac{\\sum_{k=1}^{\\infty}ka_k}{\\sum_{k=1}^{\\infty}a_k}=\\dfrac{a_1+2a_2+3a_3+&#8230;}{a_1+a_2+a_3+&#8230;}$$ Click here for the solution. Solution: For the denominator, we have: $$\\begin{align} \\sum_{n=1}^{\\infty}a_n &amp; = \\sum_{n=1}^{\\infty}\\sum_{d|n}\\dfrac{1}{2^{d+\\frac{n}{d}}} = \\sum_{d=1}^{\\infty}\\sum_{n\\ge1,d|n}\\dfrac{1}{2^{d+\\frac{n}{d}}} \\\\ &amp; = &hellip; <a href=\"https:\/\/mathfun4kids.com\/mlog\/?p=653\">Continue reading <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false},"categories":[13],"tags":[],"_links":{"self":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/653"}],"collection":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=653"}],"version-history":[{"count":48,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/653\/revisions"}],"predecessor-version":[{"id":4569,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/653\/revisions\/4569"}],"wp:attachment":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=653"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=653"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=653"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}