# Why can't MMA find the series of this function at 0?

Mathematica Asked by A little mouse on the pampas on December 4, 2020

I want to expand the following two functions into series at x = 0, but MMA(Version 12.1.1) runs all the time and cannot return results:

Series[Integrate[Log[1 + Sqrt[t^3]], {t, 0, x}], {x, 0, 5}]
Series[Integrate[Sqrt[Sin[t]^3], {t, 0, 1 - Cos[x]}], {x, 0, 5},
Assumptions -> (x > 0)]


What can I do to find their series expansion quickly?

You can use the new in M11.3 function AsymptoticIntegrate for this purpose:

AsymptoticIntegrate[Log[1+Sqrt[t^3]], {t, 0, x}, {x, 0, 5}]
AsymptoticIntegrate[Sqrt[Sin[t]^3], {t, 0, x}, {x, 0, 5}]


-(x^4/8) + (2 x Sqrt[x^3])/5 + 2/33 x^4 Sqrt[x^3]

(2 x Sqrt[x^3])/5 - 1/18 (x^3)^(3/2)

Correct answer by Carl Woll on December 4, 2020

Include Assumptions->x>0 inside Integrate

Series[Integrate[Log[1 + Sqrt[t^3]], {t, 0, x}, Assumptions -> x > 0], {x, 0, 5}]
(*(2 x^(5/2))/5 - x^4/8*)

Series[Integrate[Sqrt[Sin[t]^3], {t, 0, x}, Assumptions -> (x > 0)], {x, 0, 5}]
(*(2 x^(5/2))/5 - x^(9/2)/18*)


Answered by Ulrich Neumann on December 4, 2020

## Related Questions

### Take function as input for other function, access variable

3  Asked on May 7, 2021

### Displaying Vertex indices in different colour than vertices

1  Asked on May 7, 2021

### Minkowski sum with 2D and 3D geometries

2  Asked on May 7, 2021 by senseiwa

### Mathematica 12.x: Don’t use Accumulate or Differences on Around objects!

3  Asked on May 7, 2021

### Change dataset key name

3  Asked on May 7, 2021 by mitchell-kaplan

### Visualisation of equipotential surface – linear charge

1  Asked on May 6, 2021

### How can I find polygons in convex hull?

3  Asked on May 6, 2021 by peter-hilgers

### JuliaSetPlot – how do I achieve nicer colors?

1  Asked on May 6, 2021

### NearestNeighborGraph doesn’t show a 3D graph when one or more points are Integer and the rest are Decimal?

1  Asked on May 6, 2021 by dale-obrien

### Two values of ArcCos not showing

2  Asked on May 6, 2021

### Need help to understand the code

0  Asked on May 6, 2021

### Find the eigenvalues for 3 equations after variable separation

0  Asked on May 6, 2021 by roberta-mongillo

### How to ask Mathematica to give NumberLinePlot of $sin x$ for non-integer $x$?

2  Asked on May 6, 2021

### DSolve a partial differential equation with two boundary conditions

1  Asked on May 5, 2021

### ReplaceAll not evaluating the rule delayed function

2  Asked on May 5, 2021

### Using mesh cell highlighting to override the default rendering of a region

2  Asked on May 5, 2021 by m_goldberg

### Positions of elements in the initial flattened list in a split list

11  Asked on May 5, 2021 by garej

### Converting tensorial expressions in xAcT to LaTeX

1  Asked on May 5, 2021 by janstillerion

### Fortune’s Algorithm (Conditional Step Size in Manipulate)

0  Asked on May 5, 2021 by anirudh-bhashyam

### Plot Graphics Copy/Paste Error

0  Asked on May 5, 2021 by opticsman