Show inline math as if it were display math (and vice versa)

There are two separate aspects to your question, which can be addressed separately:

1. How to control the size of the integral, sum and product symbols

2. How to control the placement of the limits of integration, summation, and multiplication, namely side-set or above/below.

Default settings

The default settings, with the amsmath package loaded, are

• Inline math

  • Small symbol size;
  • Limits are side-set for all operators;
  • The sumlimits and intlimits options of amsmath do not affect the placement of limits in inline math.

• Display math

  • Large symbol size. See Tables 72 through 83 in the Comprehensive LaTeX Symbol List for the names of these 'large' or, more precisely, Variable-sized Math Operators;

  • For int-type symbols, the limits are side-set unless amsmath was loaded with the option intlimits.

    The integral symbols are treated separately presumably because they are generally taller than the other variable-sized symbols.

  • For sum, prod, coprod, etc the limits are set above and below the operator, except when the amsmath was loaded with the option nosumlimits.

    The sumlimits and nosumlimits options and the limits and nolimits commands affect the appearance not only of sum-type symbols but of prod,coprod, bigcup and bigcap, etc. as well.

Custom settings

• To control the size of the symbol, one writes before the command generating the symbol

  • textstyle for small symbols;
  • displaystyle for large symbols; .
  • the declarations textstyle and displaystyle may also affect the behavior of subsequent commands in the current math-mode environment, as observed by @HaraldHancheOlsen.

• To control the placement of the limits, one writes after the command generating the symbol

  • nolimits for side-set limits;
  • limits for limits set above and below.

These possibilities are illustrated in the table below:

The table was produced by the following code:

documentclass[letterpaper]{standalone}
usepackage{array,amsmath,booktabs}
begin{document}
Huge
begin{tabular}{l
        >{$textstyle}l<{$}     % first math column: text style
        >{$displaystyle}l<{$}} % second math column: display style
toprule
Placement of limits & multicolumn{2}{c}{Size of operators} 
cmidrule{2-3}
 & multicolumn{1}{l}{small:}
 & multicolumn{1}{l}{large:}
 & multicolumn{1}{l}{texttt{textbackslash textstyle}}
 & multicolumn{1}{l}{texttt{textbackslash displaystyle}}
cmidrule[lightrulewidth]{2-3}
Next to symbol: & multicolumn{1}{l}{phantom{texttt{textbackslash displaystyle }}} 
texttt{textbackslash nolimits} 
         &   sumnolimits_{i=1}^N a_i 
         &   sumnolimits_{i=1}^N a_i [2ex]
         &  prodnolimits_{j=0}^J k_j
         &  prodnolimits_{j=0}^J k_j [2.5ex]
         &   intnolimits_{-infty}^infty f(x),mathrm{d}x
         &   intnolimits_{-infty}^infty f(x),mathrm{d}x [5ex]
Below & above symbol:[-1ex]
texttt{textbackslash limits}
         &   sumlimits_{i=1}^N a_i 
         &   sumlimits_{i=1}^N a_i [3.5ex]
         &  prodlimits_{j=0}^J k_j
         &  prodlimits_{j=0}^J k_j [4ex]
         &   intlimits_{-infty}^infty f(x),mathrm{d}x
         &   intlimits_{-infty}^infty f(x),mathrm{d}x 
bottomrule
end{tabular}
end{document}

Personal comments

Finally, some personal views on the (ab)uses of the limits and displaystyle commands when in inline math mode:

• It's generally not a good idea to use the limits command when in inline math mode. Otherwise, one is virtually assured of wrecking the appearance of the paragraph where the formula appears.
• When in inline math mode, it's frequently not even necessary to indicate the full set of limits of a summation or multiplication. Expressions such as sum_i or prod_j are usually just fine. You may even be able to get away with omitting the i and j indices of summation/multiplication.
• Using the displaystyle command (to force the creation of large symbols) while in inline math mode is an even worse idea than using limits.