{"id":1286,"date":"2020-08-14T20:23:12","date_gmt":"2020-08-15T00:23:12","guid":{"rendered":"http:\/\/bln.curtisbright.com\/?p=1286"},"modified":"2020-08-14T20:23:12","modified_gmt":"2020-08-15T00:23:12","slug":"ramanujan-summation","status":"publish","type":"post","link":"http:\/\/localhost\/blog\/index.php\/2020\/08\/14\/ramanujan-summation\/","title":{"rendered":"Ramanujan summation"},"content":{"rendered":"

I just came across a way that the amazing mathematician Ramanujan<\/a> developed of assigning a value to certain divergent series. I found it interesting, so I want to share a short summary of it here. It is based on the Euler–Maclaurin formula<\/a>
\n\\begin{align*} \\sum_{k=\\alpha}^\\beta f(k) &= \\int_\\alpha^\\beta f(t)\\,dt + \\frac{f(\\alpha)+f(\\beta)}{2} \\\\ &\\quad+ \\sum_{k=1}^n \\frac{B_{2k}}{(2k)!}\\Bigl(f^{(2k-1)}(\\beta)-f^{(2k-1)}(\\alpha)\\Bigr) + R_n \\end{align*}where $B_{2k}$ denotes the $(2k)$th Bernoulli number<\/a>, $f$ has $2n+1$ continuous derivatives on $[\\alpha,\\beta]$ with $\\alpha$, $\\beta$, and $n\\geq0$ being integers, and $R_n$ is the remainder term given by
\n\\[ R_n = \\int_\\alpha^\\beta \\frac{B_{2n+1}(t-\\lfloor t\\rfloor)}{(2n+1)!}f^{(2n+1)}(t)\\,dt . \\]Taking $\\alpha=0$, $\\beta=x$ and letting $n$ tend to infinity we have (assuming that $R_n$ tends to zero)
\n\\[ \\sum_{k=0}^x f(k) \\sim \\int_a^x f(t)\\,dt + \\frac{f(x)}{2} + \\sum_{k=1}^\\infty\\frac{B_{2k}}{(2k)!}f^{(2k-1)}(x) + C(a) \\quad\\text{as $x\\to\\infty$} \\]where $C(a)$ is the constant
\n\\[ C(a) = \\int_0^a f(t)\\,dt + \\frac{f(0)}{2}-\\sum_{k=1}^\\infty\\frac{B_{2k}}{(2k)!}f^{(2k-1)}(0) . \\]Ramanujan considered $C(0)$ the “constant” of the series $\\sum f(k)$ and claimed that he considered it “like the centre of gravity of a body”.<\/p>\n

For example, letting $f(k)=1$ one sees that the series $\\sum f(k)$ diverges; however, one has $C(0)=-1\/2$ (note that all the derivatives of $f$ vanish in this case). Also, letting $f(k)=k$ one again has that $\\sum f(k)$ diverges but in this case $C(0)=-B_2\/2!=-1\/12$ as all derivatives of $f$ vanish except the first.<\/p>\n

How about a nonconverging alternating series like Grandi’s series<\/a>? Here we have $f(k)=\\cos(\\pi k)$ and in this case $C(0)=1\/2$ as $f^{(2k-1)}(0)=0$ for all $k\\geq1$ since $f^{(2k-1)}(t)=(-1)^k\\pi^{2k-1}\\sin(\\pi t)$.<\/p>\n

Conversely, consider a series that converges, e.g., when you take $f(k)=2^{-k}$. In this case one has $C(0)=1\/2+\\sum_{k=1}^\\infty B_{2k}(-\\ln2)^{2k-1}\/(2k)!\\approx0.557$—which doesn’t give the correct value of $\\sum f(k)=2$. What went wrong in this case? This is the reason for introducing the parameter $a$ above—for convergent series one needs to take $a\\to\\infty$ to achieve consistency. In this case, one computes that $\\int_0^\\infty2^{-t}\\,dt=1\/\\ln2\\approx1.443$ and one arrives at the happy result that $\\lim_{a\\to\\infty}C(a)=\\sum f(k)=2$.<\/p>\n

I’m not personally aware of any applications of this but it was too cool not to share. I found this in Chapter 6 of Ramanujan’s Notebooks, Part 1<\/a><\/i> by Bruce Berndt.<\/p>\n","protected":false},"excerpt":{"rendered":"

I just came across a way that the amazing mathematician Ramanujan developed of assigning a value to certain divergent series. I found it interesting, so I want to share a short summary of it here. It is based on the … Continue reading →<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[4],"tags":[],"_links":{"self":[{"href":"http:\/\/localhost\/blog\/index.php\/wp-json\/wp\/v2\/posts\/1286"}],"collection":[{"href":"http:\/\/localhost\/blog\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/localhost\/blog\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/localhost\/blog\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/localhost\/blog\/index.php\/wp-json\/wp\/v2\/comments?post=1286"}],"version-history":[{"count":0,"href":"http:\/\/localhost\/blog\/index.php\/wp-json\/wp\/v2\/posts\/1286\/revisions"}],"wp:attachment":[{"href":"http:\/\/localhost\/blog\/index.php\/wp-json\/wp\/v2\/media?parent=1286"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/localhost\/blog\/index.php\/wp-json\/wp\/v2\/categories?post=1286"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/localhost\/blog\/index.php\/wp-json\/wp\/v2\/tags?post=1286"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}