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Proof of p-series convergence criteria

p-series have the general form sum, start subscript, n, equals, 1, end subscript, start superscript, infinity, end superscript, start fraction, 1, divided by, n, start superscript, start superscript, p, end superscript, end superscript, end fraction where p is any positive real number. They are convergent when p, is greater than, 1 and divergent when 0, is less than, p, is less than or equal to, 1.
The AP Calculus course doesn't require knowing the proof of this fact, but we believe that as long as a proof is accessible, there's always something to learn from it. In general, it's always good to require some kind of proof or justification for the theorems you learn.
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Proof of p-series convergence criteriaSee video transcript

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  • purple pi purple style avatar for user hjy6296
    I'm at looking at the two graphs and I'm a little confused about the graph on the right side. If you add 1 to the integral, shouldn't the graph be something like a graph on the left side but the red bar on the right side graph added to it?
    (14 votes)
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    • leaf grey style avatar for user Alex
      The integral expression on the left includes the white area under the curve. The expression on the right includes the white area under the curve plus the red bar. If it's the orange series that's confusing you, it's simply because the indexes are shifted over by 1 in the graph on the right, making the red bar also belong to the orange series.
      (5 votes)
  • duskpin tree style avatar for user G Y
    At , what if you use a rectangle width other than 1?
    (3 votes)
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  • aqualine ultimate style avatar for user Chris Offner
    At don't quite understand how we see from this visualisation that 1 + Integral(...) is larger than the p-series formula in the middle. It seems like we're merely comparing it to the left Integral(...) without the +1 added to it. It makes sense to me that if we add 1 (i.e. the area of the red bar) to the area under the curve, that's bigger than the area under the curve alone, i.e. Integral(1/x^p) < 1 + Integral (1/x^p).

    But I'm not sure how we've shown that the p-series is between those two integral expressions.
    (4 votes)
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    • blobby green style avatar for user Eric Bandera
      So, the key thing is that the P series, shown in Orange are essentially left and right Riemann sums. And since the function is descending, it can be concluded that a left Riemann sum will be greater than a right Riemann sum. The integral that we are working with is from 1 to ∞. That isn't changing. So, first, the left Riemann sum from 1 to ∞. You can probably see why that is greater than the integral - there are the left over corners on the top. But, the right Riemann sum...well...we can shift the P series over to the left by one. Keep in mind the P series is equivalent to itself! We're just shifting the placement of it. But the problem is that it's not really a fair comparison after the shift since the P-series is now starting at 0 and going to ∞ instead of starting at 1 like the integral. We do know that the first term of the P series will always be 1. So...if we took the P series minus 1 would be the right Riemann series from 1 to ∞ (smaller than ∫) and the standard P series would be the left Riemann series from 1 to ∞ (bigger than ∫) This statement is: (P-series -1)< the integral < (P-series). Now keep all those inequalities in mind, and add 1 to each to get (P-series) < the integral +1 < (P-series +1). Combine them so that the integral is in the middle, and you get: the integral < (P-Series < the integral +1. This is a little convoluted, the way I explained it, but I feel after reading this the concepts will become clearer
      (6 votes)
  • starky tree style avatar for user {Rayeed}^3
    I understand it fully in mathematical terms but I don't have a nice intuition about it. If we look at the graphs of 1/x and 1/x^2, they both look almost the same. The fact that one of them has a finite area and the other has an infinite area seems counterintuitive. Do both of them 'touch' the x axis as the reach infinity? How does all this actually happen?
    (6 votes)
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    • leaf green style avatar for user CJ Coudriet
      My intuition for this is somewhat mentioned in the comment below. Basically, the terms in the sequence decrease towards 0 at a slower and slower rate, allowing the sum to approach infinity. Have you ever looked at the graph of ln(x)? If you look on desmos (https://www.desmos.com/calculator/iing9fbgoc) and slowly scroll to the right, it feels like the values keep growing by smaller and smaller values, but nevertheless always growing. It makes sense that the graph of ln(x) must approach infinity since you can plug any number into e^x (the "answer" for ln(x)). Then, zoom out in the graph and go as far right as you'd like. The numbers are much larger, and they never seem to stop like convergent series do, even though it grows so slowly. Even if that didn't make any sense, think about this: the difference between any two consecutive values doesn't change very fast, so the sum is able to continually grow. Reasoning from Ian Pulizzotto (thanks!):

      "1+1/2+1/3+1/4+1/5+1/6+1/7+1/8+1/9+1/10+1/11+1/12+1/13+1/14+1/15+1/16+...

      = 1+1/2+(1/3+1/4)+(1/5+1/6+1/7+1/8)+(1/9+1/10+1/11+1/12+1/13+1/14+1/15+1/16)+...

      >= 1+1/2+(1/4+1/4)+(1/8+1/8+1/8+1/8)+(1/16+1/16+1/16+1/16+1/16+1/16+1/16+1/16)+...

      = 1+1/2+2/4+4/8+8/16+...

      = 1+1/2+1/2+1/2+1/2+..., which clearly diverges to infinity since the sequence 1,1.5,2,2.5,3,... clearly grows without bound."

      This doesn't work with 1/x^2, for instance:
      1/1+1/4+1/9+1/16+1/25+...
      > 1+(1/9+1/9)+(1/25+1/25)
      which doesn't diverge to anything. It simply gives another sequence. How I think about it is that 1/x^n, where n > 1, has the denominator of 1/k skipping more and more values (1, 1/4, 1/9,1/16, 1/25, 1/36, etc.) while 1/x passes through every value which sums up to infinity.

      I'm not sure though, bc not I'm more confused after writing this comment.

      EDIT: Read my comment below! It has more examples!
      (1 vote)
  • sneak peak green style avatar for user G. Tarun
    What does "divergent" mean in the context of a p-series?
    Does it mean that as the series goes on—endlessly, toward infinity—the terms get larger and larger?
    If p=1, then the the p-series is divergent by definition, as a divergent p-series has a value of p greater than zero but lesser than or equal to 1 (as given in this article and the Harmonic series and p-series video in this lesson). But then, in a harmonic p-series whose p value is 1, don't the terms get smaller and smaller as the series goes on?
    (1 vote)
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    • primosaur seed style avatar for user Ian Pulizzotto
      Divergence of a series does not always mean that the terms get larger or stay the same size. Divergence of a series means that the limit of the partial sums of the terms does not exist (that is, the partial sums grow without bound, positively or negatively, or the partial sums oscillate without converging to a limit).

      In the case of the harmonic series with p=1, yes the terms do become smaller and converge to zero, but the terms converge so slowly to zero that the partial sums still grow without bound! We can see this by observing that

      1+1/2+1/3+1/4+1/5+1/6+1/7+1/8+1/9+1/10+1/11+1/12+1/13+1/14+1/15+1/16+...

      = 1+1/2+(1/3+1/4)+(1/5+1/6+1/7+1/8)+(1/9+1/10+1/11+1/12+1/13+1/14+1/15+1/16)+...

      >= 1+1/2+(1/4+1/4)+(1/8+1/8+1/8+1/8)+(1/16+1/16+1/16+1/16+1/16+1/16+1/16+1/16)+...

      = 1+1/2+2/4+4/8+8/16+...

      = 1+1/2+1/2+1/2+1/2+..., which clearly diverges to infinity since the sequence 1,1.5,2,2.5,3,... clearly grows without bound.

      So the harmonic series with p=1 diverges to infinity!

      It is important the distinguish the behavior of the sequence of terms from the behavior of the partial sums of the terms, since these behaviors are not always the same.
      (9 votes)
  • blobby green style avatar for user Soldier King
    at , how do you know p is greater than 0?
    (1 vote)
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  • starky ultimate style avatar for user Charizard-Max
    why is the sum of 1/n^p with 0<p<1 diverging on using m->infinity(using lim) but on the graph literally shows that as n-> infinity 1/n^p->0 (0<p<1)(even if it's slow but infinity is a really big value, in the end it still converge to zero for 1/n^p)?

    why is that?
    (3 votes)
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    • blobby green style avatar for user Wantedsolo
      This is quite a late answer but ig it's for future reference..

      I think you are a bit confused between divergence of series and divengence of nth term:
      the final term of 1/n^p for any non negative value of p will always approach 0 as lim n -> infinity. But that says nothing about the sum of the series.

      The y values of the graph itself only show the individual values of 1/n^p at different points. But the sum of the series (and what we're trying to calculate) is the area of the rieman sums of the graph and thus bounded on both sides by the 2 graphs. These sums are not necessarily finite and the conditions that change the sums 'finiteness' is what was proved in this video.

      Hope this resolved your doubt.
      (1 vote)
  • starky ultimate style avatar for user Michele Franzoni
    I've read all the answers and comments in this section but i can't really grasp the reason why the series is bounded the way it is.

    If the series is greater than the improper integral but lesser than the improper integral plus one it means that such integral overestimates the series by less than one (when looking to the let graph). And the reason why this should be true is very far from clear. Could someone help me out?
    (2 votes)
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    • male robot hal style avatar for user JI YONG Ahn
      1) Integration of P-series from 1 to infinity is the white shade.
      2) The summation of the P-series from 1 to infinity is the bars. In comparison to clause 1), it has larger surface area.
      3) Integration of P-series from 1 to infinity + 1 is the red and white area.
      4) now, see the left graph and right graph. They both have the bars. The bars are the summation of the P-series in both cases.
      5) the right side graph is the left side bars moved to the left by 1. This is done by doing +1.

      Therefore, "such integral overestimates" as you mentioned, because it is, by looking at it. It is the surface areas of the bars + the gap between the function's curve and the bars.

      As a side note:
      1) the area of overestimation on the left graph is, (the summation of P series) - (the integration of the P series). This is the overestimated area of the Summation of P-series compared to the area by integration.
      2) the area of underestimation on the right graph, for the Summation of P-series is, (1 + integration of P series) - (summation of P series).

      1) and 2) are different.
      (3 votes)
  • orange juice squid orange style avatar for user Nico Ziegler
    How is it about p=0?
    Does it simplify to infinit sum of 1+1+..+1 which diverges?
    (1 vote)
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  • blobby green style avatar for user Hexuan Sun 9th grade
    What will happen if the p exponent is less than 0?
    (1 vote)
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